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Dynamical Analogies
In the manufacture of this book, the publishers
have observed the recommendations of the WarProduction Board with respect to paper, printing
and binding in an effort to aid in the conserva-
tion of paper and other critical war materials.
Dynamical Analogies
By
HARRY F. OLSON, E.E., Ph.D.
A coustical Research Director
RCA LaboratorieSy Princeton, New Jersey
NEW YORK
D. VAN NOSTRAND COMPANY, Inc.
250 Fourth Avenue
1943
Copyright, 1943, by
D. VAN NOSTRAND COMPANY, Inc.
All Rights Reserved
This book, or any -parts thereof, maynot he reprodii-ced in any form without
written permission from the publishers.
Printed in U. S. A.
PREFACE
Analogies are useful for analysis in unexplored fields. By means of
analogies an unfamiliar system may be compared with one that is better
known. The relations and actions are more easily visualized, the mathe-
matics more readily applied and the analytical solutions more readily
obtained in the familiar system.
Although not generally so considered the electrical circuit is the most
common and widely exploited vibrating system. By means of analogies
the knowledge in electrical circuits may be applied to the solution of
problems in mechanical and acoustical systems. In this procedure the
mechanical or acoustical vibrating system is converted into the analogous
electrical circuit. The problem is then reduced to the simple solution of
an electrical circuit. This method has been used by acoustical engineers
for the past twenty years in the development of all types of electro-
acoustic transducers. Mechanical engineers have begun to use the sameprocedure for analyzing the action of mechanisms.
The importance and value of dynamical analogies to any one con-
cerned with vibrating systems have led to a demand for expositions on
this branch of dynamics. Accordingly this book has been written with
the object of presenting the principles of dynamical analogies to the
engineer.
This book deals with the analogies between electrical, mechanical
rectilineal, mechanical rotational and acoustical systems. The subject
matter is developed in stages from the simple element through to com-plex arrangements of multielement systems. As an aid in the establish-
ment of these analogies a complete theme is depicted in each illustration.
The text assumes on the part of the reader a familiarity with the ele-
ments of alternating circuit theory and physics.
The author wishes to express his gratitude to his wife, Lorene E. Olson,
for compilation and assistance in preparation and correction of the
manuscript.
The author wishes to acknowledge the interest given by Mr. E. W.Engstrom, Research Director, in this project.
Harry F. OlsonJanuary, 1943
CONTENTS
Chapter Page
I. INTRODUCTION AND DEFINITIONS
1.1 Introduction 1
1.2 Definitions 4
II. ELEMENTS
2.1 Introduction 12
2.2 Resistance 12
A. Electrical Resistance 12
B. Mechanical Rectilineal Resistance 13
C. Mechanical Rotational Resistance 13
D. Acoustical Resistance 13
2.3 Inductance, Mass, Moment of Inertia, Inertance 15
A. Inductance 15
B. Mass 15
C. Moment of Inertia 15
D. Inertance 16
2.4 Electrical Capacitance, Rectilineal Compliance, Rotational
Compliance, Acoustical Capacitance 17
A. Electrical Capacitance 17
B. Rectilineal Compliance 17
C. Rotational Compliance 18
D. Acoustical Capacitance 18
2.5 Representation of Electrical, Mechanical Rectilineal, Me-chanical Rotational and Acoustical Elements 19
III. ELECTRICAL, MECHANICAL RECTILINEAL, MECHANICAL RO-TATIONAL, AND ACOUSTICAL SYSTEMS OF ONE DEGREE OFFREEDOM
3.1 Introduction 25
3.2 Description of Systems of One Degree of Freedom 25
3.3 Kinetic Energy 27
3.4 Potential Energy 28
vii
viii CONTENTSChapter Page
3.5 Dissipation 29
3.6 Equations of Motion 30
3.7 Resonant Frequency 32
3.8 Kirchhoff's Law and D'Alembert's Principle 33
IV. ELECTRICAL, MECHANICAL RECTILINEAL, MECHANICAL RO-TATIONAL AND ACOUSTICAL SYSTEMS OF TWO AND THREEDEGREES OF FREEDOM
4.1 Introduction 37
4.2 Two Degrees of Freedom 37
4.3 Kinetic Energy 38
4.4 Potential Energy 39
4.5 Dissipation 39
4.6 Equations of Motion 40
4.7 The Electrical System , 41
4.8 The Mechanical Rectilineal System 41
4.9 The Mechanical Rotational System 42
4.10 The Acoustical System 42
4.11 Comparison of the Four Systems 43
4.12 Electrical Inductive and Capacitive Coupled Systems of TwoDegrees of Freedom and the Mechanical Rectilineal, Me-chanical Rotational and Acoustical Analogies 45
4.13 Electrical, Mechanical Rectilineal, Mechanical Rotational
and Acoustical Systems of Three Degrees of Freedom 48
V. CORRECTIVE NETWORKS
5.1 Introduction 52
5.2 Two Electrical, Mechanical Rectilineal, Mechanical Rota-
tional or Acoustical Impedances in Parallel 52
5.3 Shunt Corrective Networks 56
5.4 Inductance in Shunt with a Line and the Mechanical Recti-
lineal, Mechanical Rotational and Acoustical Analogies .... 58
5.5 Electrical Capacitance in Shunt with a Line and the Me-chanical Rectilineal, Mechanical Rotational and Acoustical
Analogies 60
5.6 Inductance and Electrical Capacitance in Series, in Shuntwith a Line and the Mechanical Rectilineal, MechanicalRotational and Acoustical Analogies 62
5.7 Inductance and Electrical Capacitance in Parallel, in Shunt
with a Line and the Mechanical Rectilineal, MechanicalRotational and Acoustical Analogies 64
5.8 Electrical Resistance, Inductance and Electrical Capacitance
IN Series, in Shunt with a Line and the Mechanical Recti-
lineal, Mechanical Rotational and Acoustical Analogies. ... 67
CONTENTS ix
Chapter Page
5.9 Electrical Resistance, Inductance and Electrical Capaci-
tance IN Parallel, in Shunt with a Line and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analo-
gies 69
5.10 Series Corrective Networks 71
5.11 Inductance in Series with a Line and the Mechanical Recti-
lineal, Mechanical Rotational and Acoustical Analogies .... 72
5.12 Electrical Capacitance in Series with a Line and the Me-chanical Rectilineal, Mechanical Rotational and Acoustical
,'\nalog:es 74
5.13 Inductance and Electrical Capacitance in Series with a Line
and the Mechanical Rectilineal, Mechanical Rotational, and
Acoustical Analogies 76
5.14 Inductance and Electrical Capacitance in Parallel, in Series
with a Line and the Mechanical Rectilineal, Mechanical
Rotational and Acoustical Analogies 78
5.15 Electrical Resistance, Inductance and Electrical Capacitance
IN Series with a Line and the Mechanical Rectilineal, Me-chanical Rotational and Acoustical Analogies 80
5.16 Electrical Resistance, Inductance and Electrical Capaci-
tance IN Parallel, in Series with a Line and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies 83
5.17 Resistance Networks 85
5.18 Electrical Resistance in Series with a Line and the Mechan-ical Rectilineal, Mechanical Rotational, and Acoustical Anal-
ogies 85
5.19 Electrical Resistance in Shunt with a Line and the Mechan-
ical Rectilineal, Mechanical Rotational and Acoustical Anal-
ogies 86
5.20 "T" Type Electrical Resistance Network and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies 87
5.21 "it" Type Electrical Resistance Network and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies 87
5.22 Electrical, Mechanical Rectilineal, Mechanical Rotational
and Acoustical Transformers 88
VI. WAVE FILTERS
6.1 Introduction 92
6.2 Types of Wave Filters 92
6.3 Response Characteristics of Wave Filters 93
6.4 Low Pass Wave Filters 94
6.5 High Pass Wave Filters 95
6.6 Band Pass Wave Filters 97
6.7 Band Elimination Wave Filters 101
X CONTENTSChapter PageVII. TRANSIENTS
7. 1 Introduction lOS
7.2 The Heaviside Operational Calculus 106
7.3 Transient Response of an Inductance and Electrical Resist-
ance IN Series and the Mechanical Rectilineal, Mechanical
Rotational and Acoustical Analogies 107
7.4 Transient Response of an Electrical Resistance and Elec-
trical Capacitance in Series and the Mechanical Rectilineal,
Mechanical Rotational and Acoustical Analogies Ill
7.5 Transient Response of an Electrical Resistance, Inductance
AND Electrical Capacitance in Series and the MechanicalRectilineal, Mechanical Rotational and Acoustical Analogies 114
7.6 Arbitrary Force 120
VIII. DRIVING SYSTEMS
8.1 Introduction 124
8.2 Electrodynamic Driving System 124
8.3 Electromagnetic Driving Systems 126
A. Unpolarized Armature Type 127
B. Polarized Reed Armature Type 130
C. Polarized Balanced Armature Type 134
8.4 Electrostatic Driving System 138
8.5 Magnetostriction Driving Sy.stem 141
8.6 Piezoelectric Driving System 148
IX. GENERATING SYSTEMS
9.1 Introduction 153
9.2 Electrodynamic Generating System 153
9.3 Electromagnetic Generating Systems 155
A. Reed Armature Generating System 155
B. Balanced Armature Generating System 157
9.4 Electrostatic Generating System 158
9.5 Magnetostriction Generating System 162
9.6 Piezoelectric Generating System 165
X. THEOREMS
10.1 Introduction 171
10.2 Reciprocity Theorems 171
A. Electrical Reciprocity Theorem 171
B. Mechanical Rectilineal Reciprocity Theorem 172
C. Mechanical Rotational Reciprocity Theorem 173
D. Acoustical Reciprocity Theorem 173
CONTENTS xi
Chapter PageE. Mechanical-Acoustical Reciprocity Theorem 175
F. Electrical-Mechanical Reciprocity Theorem 176
G. Electrical-Mechanical-Acoustical Reciprocity Theorem 177
H. Electrical-Mechanical-Acoustical-Mechanical-Elcctrical Reci-
procity Theorem 177
I. Acoustical-Mcchanical-Electrical-Mechanical-Acoustical Reci-
procity Theorem 178
10.3 Thevenin's Theorems 178
A. Thevenin's Electrical Theorem 178
B. Thevenin's Mechanical Rectilineal Theorem 178
C. Thevenin's Mechanical Rotational Theorem 178
D. Thevenin's Acoustical Theorem 178
10.4 Superposition Theorem 179
XI. APPLICATIONS
11.1 Introduction 180
11.2 Automobile Muffler 180
11.3 Electric Clipper 182
11.4 Direct Radiator Loud Speaker 183
11.5 Rotational Vibration Damper 184
11.6 Machine Vibration Isolator 185
11.7 Mechanical Refrigerator Vibration Isolator 186
11.8 Shockproof Instrument Mounting 187
1 1 .9 Automobile Suspension System 188
INDEX 191
CHAPTER I
INTRODUCTION AND DEFINITIONS
1.1. Introduction
Analogies are useful when it is desired to compare an unfamiliar system
with one that is better known. The relations and actions are more easily
visualized, the mathematics more readily applied and the analytical
solutions more readily obtained in the familiar system. Analogies makeit possible to extend the line of reasoning into unexplored fields.
A large part of engineering analysis is concerned with vibrating sys-
tems. Although not generally so considered, the electrical circuit is the
most common example and the most widely exploited vibrating system.
The equations of electrical circuit theory may be based on Maxwell's
dynamical theory in which the currents play the role of velocities.
Expressions for the kinetic energy, potential energy and dissipation show
that network equations are deducible from general dynamic equations.
In other words, an electrical circuit may be considered to be a vibrating
system. This immediately suggests analogies between electrical circuits
and other dynamical systems, as for example, mechanical and acoustical
vibrating systems.
The equations of motion of mechanical systems were developed a long
time before any attention was given to equations for electrical circuits.
For this reason, in the early days of electrical circuit theory, it was nat-
ural to explain the action in terms of mechanical phenomena. However,
at the present time electrical circuit theory has been developed to a
much higher state than the corresponding theory of mechanical sys-
tems. The number of engineers and scientists versed in electrical
circuit theory is many times the number equally familiar with mechanical
systems.
Almost any work involving mechanical or acoustical systems also
includes electrical systems and electrical circuit theory. The acoustical
1
2 INTRODUCTION AND DEFINITIONS
engineer is interested in sound reproduction or the conversion of electrical
or mechanical energy into acoustical energy, the development of vibrat-
ing systems and the control of sound vibrations. This involves acousti-
cal, electroacoustical, mechanoacoustical or electromechanoacoustical
systems. The mechanical engineer is interested in the development of
various mechanisms or vibrating systems involving masses, springs and
friction.
Electrical circuit theory is the branch of electromagnetic theory which
deals with electrical oscillations in linear electrical networks.^ An elec-
trical network is a connected set of separate circuits termed branches or
meshes. A circuit may be defined as a physical entity in which varying
magnitudes may be specified in terms of time and a single dimension.
^
The branches or meshes are composed of elements. Elements are the
constituent parts of a circuit. Electrical elements are resistance, induct-
ance and capacitance. Vibrations in one dimension occur in mechanical
systems made up of mechanical elements, as for example, various assem-
blies of masses, springs and brakes. Acoustical systems in which the
dimensions are small compared to the wavelength are vibrations in a
single dimension.
The number of independent variables required to completely specify
the motion of every part of a vibrating system is a measure of the number
of degrees of freedom of the system. If only a single variable is needed
the system is said to have a single degree of freedom. In an electrical
circuit the number of degrees of freedom is equal to the number of inde-
pendent closed meshes or circuits.
The use of complex notation has been applied extensively to electrical
circuits. Of course, this operational method can be applied to any
analytically similar system.
Mathematically the elements in an electrical network are the coeffi-
cients in the differential equations describing the network. When the
electric circuit theory is based upon Maxwell's dynamics the network
forms a dynamical system in which the currents play the role of veloci-
ties. In the same way the coefficients in the differential equations of
' The use of the terms "circuit" and "network" in the literature is not estab-
lished. The term "circuit" is often used to designate a network with several
branches.^ The term "single dimension" implies that the movement or variation occurs
along a path. In a field problem there is variation in two or three dimensions.
INTRODUCTION 3
a mechanical or acoustical system may be looked upon as mechanical or
acoustical elements. Kirchhoff's electromotive force law plays the same
role in setting up the electrical equations as D'Alembert's principle does
in setting up the mechanical and acoustical equations. That is to say,
every electrical, mechanical or acoustical system may be considered
as a combination of electrical, mechanical or acoustical elements.
Therefore, any mechanical or acoustical system may be reduced to an
electrical network and the problem may be solved by electrical circuit
theory.
In view of the tremendous amount of study which has been directed
towards the solution of circuits, particularly electrical circuits, and the
engineer's familiarity with electrical circuits, it is logical to apply this
knowledge to the solution of vibration problems in other fields by the
same theory as that used in the solution of electrical circuits.
In this book, the author has attempted to outline the essentials of
dynamical analogies ' from the standpoint of the engineer or applied
scientist. Differential equations are used to show the basis for the
analogies between electrical, mechanical and acoustical systems. How-ever, the text has been written and illustrated so that the derivations
may be taken for granted. The principal objective in this book is the
establishment of analogies between electrical, mechanical and acoustical
systems so that any one familiar with electrical circuits will be able to
analyze the action of vibrating systems.
^ The analogies as outlined in this book are formal ones due to the similarity
of the differential equations and do not imply that there is any physical similar-
ity between quantities occvipying the same position in their respective equations.
There is no claim that the analogies as outlined in this book are the only onespossible. For example, in the past, mechanical impedance has been defined bysome authors as the ratio of pressure to velocity, ratio of force to displacement,and ratio of pressure to displacement. Hanle {JViss. Verojf a. d. Siemens-Konzern, Vol. XI, No. I) and Firestone {Jour. Acous. Soc. Amer., Vol. 4, No. 4,
1933) have proposed analogies in which mechanical impedance is defined as theratio of velocity to force. Every analogy possesses certain advantages, par-
ticularly in the solution of certain specific problems. However, the analogiesas defined in this book conform with the American Standard Acoustical Termi-nology—z24.1 of 1942; and the Standards of Electroacoustics, Institute ofRadio Engineers. In addition, all communication, circuit, and electrical engi-
neering books employing analogies to explain alternating current phenomenause analogies as defined in this book. Finally, analogies as defined in this bookare universally employed in the technical and scientific journals. Therefore, it
is only logical to conform with the recognized standards and preponderance ofusage.
4 INTRODUCTION AND DEFINITIONS
1.2. Definitions
A few of the terms ^ used in dynamical analogies will be defined in
this section. Terms not listed below will be defined in subsequent
sections.
Periodic Quantity.—A periodic quantity is an oscillating quantity the
values of which recur for equal increments of the independent variable.
If a periodic quantity jy is a function of ^, then j has the property that
y — f{^) = f(x + T), where T, a constant, is a period of jy. The small-
est positive value of T'ls the primitive period ofy, generally called simply
the period of jy. In general a periodic function can be expanded into a
series of the form.
y = /('') = ^0 + ^1 sin (co^ + ai) + ^2 sin (2iox + 0:2) + • • • j
where co, a positive constant, equals 2x divided by the period T, and the
yf's and a's are constants which may be positive, negative, or zero. This
is called a Fourier series.
Cycle.—One complete set of the recurrent values of a periodic quan-
tity comprises a cycle.
Period.—The time required for one cycle of a periodic quantity is the
period. The unit is the second.
Frequency.—The number of cycles occurring per unit of time, or which
would occur per unit of time if all subsequent cycles were identical with
the cycle under consideration, is the frequency. The frequency is the
reciprocal of the period. The unit is the cycle per second.
Octave.—An octave is the interval between two frequencies having a
ratio of two to one.
Fundamental Frequency
.
—A fundamental frequency is the lowest com-
ponent frequency of a periodic quantity.
Harmonic.—A harmonic is a component of a periodic quantity which
is an integral multiple of the fundamental frequency. For example, a
component the frequency of which is twice the fundamental frequency is
called the second harmonic.
Basic Frequency.—The basic frequency of a periodic quantity is that
frequency which is considered to be the most important. In a driven
^ Approximately one-half of the definitions in this chapter are taken from the
American Standards Association standards. The remainder, which have not
been defined at this time by any standards group, are written to conform with
the analogous existing standards.
DEFINITIONS 5
system it would in general be the driving frequency while in most
periodic waves it would correspond to the fundamental frequency.
Subharmonic.—A subharmonic is a component of a periodic quantity
having a frequency which is an integral submultiple of the basic fre-
quency.
Note: The term "subharmonic" is generally applied in the case of a
driven system whose vibration has frequency components of lower fre-
quency than the driving frequency.
Wave.—A wave is a propagated disturbance, usually a periodic quan-
tity in an electrical, mechanical or acoustical system.
Wavelength.—The wavelength of a periodic wave in an isotropic
medium is the perpendicular distance between two wave fronts in which
the displacements have a phase difference of one complete cycle.
Abvolt.—An abvolt is the unit of electromotive force.
Instantaneous Electromotive Force.—The instantaneous electromotive
force between two points is the total instantaneous electromotive force.
The unit is the abvolt.
Effective Electromotive Force.—The effective electromotive force is the
root mean square of the instantaneous electromotive force over a com-
plete cycle between two points. The unit is the abvolt.
Maximum Electromotive Force.—The maximum electromotive force is
the maximum absolute value of the instantaneous electromotive force
during that cycle. The unit is the abvolt.
Peak Electromotive Force.—The peak electromotive force for any speci-
fied time interval is the maximum absolute value of the instantaneous
electromotive force during that cycle. The unit is the abvolt.
Dyne.—A dyne is the unit of force or mechanomotive force.
Instantaneous Force {Instantaneous Mechanomotive Force).—The in-
stantaneous force at a point is the total instantaneous force. The unit is
the dyne.
Effective Force {Effective Mechanomotive Force).—The effective force is
the root mean square of the instantaneous force over a complete cycle.
The unit is the dyne.
Maximum Force {Maximum Mechanomotive Force).—The maximumforce is the maximum absolute value of the instantaneous force during
that cycle. The unit is the dyne.
Peak Force {Peak Mechanomotive Force).—The peak force for any
specified interval is the maximum absolute value of the instantaneous
force during that cycle. The unit is the dyne.
6 INTRODUCTION AND DEFINITIONS
Dyne Centimeter.—A dyne centimeter is the unit of torque or rotato-
motive force.
Instantaneous Torque {Instantaneous Rotatomotive Force).—The instan-
taneous torque at a point is the total instantaneous torque. The unit is
the dyne centimeter.
Effective Torque {Effective Rotatomotive Force).—The effective torque is
the root mean square of the instantaneous torque over a complete cycle.
The unit is the dyne centimeter.
Maximum Torque {Maximum Rotatomotive Force).—The maximumtorque is the maximum absolute value of the instantaneous torque during
that cycle. The unit is the dyne centimeter.
Peak Torque {Peak Rotatomotive Force)
.
—The peak torque for a speci-
fied interval is the maximum absolute value of the instantaneous torque
during that cycle. The unit is the dyne centimeter.
Dyne per Square Centimeter.—A dyne per square centimeter is the unit
of sound pressure or acoustomotive force.
Static Pressure.—The static pressure is the pressure that would exist
in a medium with no sound waves present. The unit is the dyne per
square centimeter.
Instantaneous Sound Pressure {Instantaneous Acoustomotive Force).—The instantaneous sound pressure at a point is the total instantaneous
pressure at the point minus the static pressure. The unit is the dyne per
square centimeter.
Effective Sound Pressure {Effective Acoustomotive Force).—The effective
sound pressure at a point is the root mean square value of the instantane-
ous sound pressure over a complete cycle at the point. The unit is the
dyne per square centimeter.
Maximum Sound Pressure {Maximum Acoustomotive Force).—Themaximum sound pressure for any given cycle is the maximum absolute
value of the instantaneous sound pressure during that cycle. The unit is
the dyne per square centimeter.
Peak Sound Pressure {Peak Acoustomotive Force).—The peak sound
pressure for any specified time interval is the maximum absolute value
of the instantaneous sound pressure in that interval. The unit is the
dyne per square centimeter.
Abampere.—An abampere is the unit of current.
Instantaneous Current.—The instantaneous current at a point is the
total instantaneous current at that point. The unit is the abampere.
DEFINITIONS 7
Effective Current.—The effective current at a point is the root meansquare value of the instantaneous current over a complete cycle at that
point. The unit is the abampere.
Maximum Current.—The maximum current for any given cycle is the
maximum absolute value of the instantaneous current during that cycle.
The unit is the abampere.
Peak Current.—The peak current for any specified time interval is the
maximum absolute value of the instantaneous current in that interval.
The unit is the abampere.
Centimeter per Second.—A centimeter per second is the unit of velocity.
Instantaneous Velocity.—The instantaneous velocity at a point is the
total instantaneous velocity at that point. The unit is the centimeter
per second.
Effective Velocity.—The effective velocity at a point is the root meansquare value of the instantaneous velocity over a complete cycle at that
point. The unit is the centimeter per second.
Maximum Velocity.—The maximum velocity for any given cycle is the
maximum absolute value of the instantaneous velocity during that cycle.
The unit is the centimeter per second.
Peak Velocity.—The peak velocity for any specified time interval is the
maximum absolute value of the instantaneous velocity in that interval.
The unit is the centimeter per second.
Radian per Second.—A radian per second is the unit of angular
velocity.
Instantaneous Angular Velocity.—The instantaneous angular velocity
at a point is the total instantaneous angular velocity at that point. Theunit is the radian per second.
Effective Angular Velocity.—The effective angular velocity at a point is
the root mean square value of the instantaneous angular velocity over a
complete cycle at the point. The unit is the radian per second.
Maximum Angular Velocity.—The maximum angular velocity for any
given cycle is the maximum absolute value of the instantaneous angular
velocity during that cycle. The unit is the radian per second.
Peak Angular Velocity.—The peak angular velocity for any specified
time interval is the maximum absolute value of the instantaneous angular
velocity in that interval. The unit is the radian per second.
Cubic Centimeter per Second.—A cubic centimeter is the unit of volume
current.
8 INTRODUCTION AND DEFINITIONS
Instantaneous Volume Current.—The instantaneous volume current at
a point is the total instantaneous volume current at that point. The unit
is the cubic centimeter per second.
Effective Volume Current.—The effective volume current at a point is
the root mean square value of the instantaneous volume current over a
complete cycle at that point. The unit is the cubic centimeter per
second.
Maximum Volume Current.—The maximum volume current for any
given cycle is the maximum absolute value of the instantaneous volume
current during that cycle. The unit is the cubic centimeter per second.
Peak Volume Current.—The peak volume current for any specified
time interval is the maximum absolute value of the instantaneous volume
current in that interval. The unit is the cubic centimeter per second.
Electrical Impedance.—Electrical impedance is the complex quotient
of the alternating electromotive force applied to the system by the result-
ing current. The unit is the abohm.
Electrical Resistance.—Electrical resistance is the real part of the elec-
trical impedance. This is the part responsible for the dissipation of
energy. The unit is the abohm.
Electrical Reactance.—Electrical reactance is the imaginary part of the
electrical impedance. The unit is the abohm.
Inductance.—Inductance in an electrical system is that coefficient
which, when multiplied by 2.-W times the frequency, gives the positive
imaginary part of the electrical impedance. The unit is the abhenry.
Electrical Capacitance.—Electrical capacitance in an electrical system
is that coefficient which, when multiplied by 1-k times the frequency, is
the reciprocal of the negative imaginary part of the electrical impedance.
The unit is the abfarad.
Mechanical RectilinealImpedance^ {MechanicalImpedance)
.
—Mechan-ical rectilineal impedance is the complex quotient of the alternating force
applied to the system by the resulting linear velocity in the direction of
the force at its point of application. The unit is the mechanical ohm.
^The word "mechanical" is ordinarily used as a modifier to designate amechanical system with rectilineal displacements and the word "rotational" is
ordinarily used as a modifier to designate a mechanical system with rotational
displacements. To avoid ambiguity in this book, where both systems are con-sidered concurrently, the words "mechanical rectilineal" are used as modifiers
to designate a mechanical system with rectilineal displacements and the words"mechanical rotational" are used as modifiers to designate a mechanical systemwith rotational displacements.
DEFINITIONS 9
Mechanical Rectilineal Resistance {Mechanical Resistance^.—Mechani-
cal rectilineal resistance is the real part of the mechanical rectilineal
impedance. This is the part responsible for the dissipation of energy.
The unit is the mechanical ohm.
Mechanical Rectilineal Reactance {Mechanical Reactance)
.
—Mechanical
rectilineal reactance is the imaginary part of the mechanical rectilineal
impedance. The unit is the mechanical ohm.
Mass.—Mass in a mechanical system is that coefficient which, whenmultiplied by liv times the frequency, gives the positive imaginary part
of the mechanical rectilineal impedance. The unit is the gram.
Compliance.—Compliance in a mechanical system is that coefficient
which, when multiplied by 1-k times the frequency, is the reciprocal of
the negative imaginary part of the mechanical rectilineal impedance.
The unit is the centimeter per dyne.
Mechanical Rotational Impedance ^ {RotationalImpedance)
.
—Mechani-
cal rotational impedance is the complex quotient of the alternating torque
applied to the system by the resulting angular velocity in the direction of
the torque at its point of application. The unit is the rotational ohm.Mechanical Rotational Resistance {Rotational Resistance),—Mechanical
rotational resistance is the real part of the mechanical rotational imped-
ance. This is the part responsible for the dissipation of energy. Theunit is the rotational ohm.
Mechanical Rotational Reactance {Rotational Reactance).—Mechanical
rotational reactance is the imaginary part of the mechanical rotational
impedance. The unit is the rotational ohm.
Moment of Inertia.—Moment of inertia in a mechanical rotational
system is that coefficient which, when multiplied by lir times the fre-
quency, gives the positive imaginary part of the mechanical rotational
impedance. The unit is the gram centimeter to the second power.
Rotational Compliance.—Rotational compliance in a mechanical rota-
tional system is that coefficient which, when multiplied by 1-k times the
frequency, is the reciprocal of the negative imaginary part of the mechan-
ical rotational impedance. The unit is the radian per centimeter per
dyne.
Acoustical Impedance.—Acoustical Impedance is the complex quotient
of the pressure applied to the system by the resulting volume current.
The unit is the acoustical ohm.
' See footnote S, page I
10 INTRODUCTION AND DEFINITIONS
Acoustical Resistance.—Acoustical resistance is the real part of the
acoustical impedance. This is the part responsible for the dissipation of
energy. The unit is the acoustical ohm.
Acoustical Reactance.—Acoustical reactance is the imaginary part of
the acoustical impedance. The unit is the acoustical ohm.
Inertance.—Inertance in an acoustical system is that coefficient which,
when multiplied by Itt times the frequency, gives the positive imaginary
part of the acoustical impedance. The unit is the gram per centimeter to
the fourth power.
Acoustical Capacitance.—Acoustical capacitance in an acoustical sys-
tem is that coefficient which, when multiplied by "l-w times the frequency,
is the reciprocal negative imaginary part of the acoustical impedance.
The unit is the centimeter to the fifth power per dyne.
Element?—An element or circuit parameter in an electrical system
defines a distinct activity in its part of the circuit. In the same way, an
element in a mechanical rectilineal, mechanical rotational or acoustical
system defines a distinct activity in its part of the system. The elements
in an electrical circuit are electrical resistance, inductance and electrical
capacitance. The elements in a mechanical rectilineal system are
mechanical rectilineal resistance, mass and compliance. The elements in
a mechanical rotational system are mechanical rotational resistance,
moment of inertia, and rotational compliance. The elements in an
acoustical system are acoustical resistance, inertance and acoustical
capacitance.
Electrical System.—An electrical system is a system adapted for the
transmission of electrical currents consisting of one or all of the electrical
elements: electrical resistance, inductance and electrical capacitance.
Mechanical Rectilineal System.—A mechanical rectilineal system is a
system adapted for the transmission of linear vibrations consisting of one
or all of the following mechanical rectilineal elements: mechanical recti-
lineal resistance, mass and compliance.
Mechanical Rotational System,.—A mechanical rotational system is a
system adapted for the transmission of rotational vibrations consisting of
one or all of the following mechanical rotational elements: mechanical
rotational resistance, moment of inertia and rotational compliance.
Acoustical System.—An acoustical system is a system adapted for the
transmission of sound consisting of one or all of the following acoustical
elements: acoustical resistance, inertance and acoustical capacitance.
^ Elements are defined and described in Chapter II.
DEFINITIONS 11
Transducer.—A transducer is a device actuated by power from one
system and supplying power in the same or any other form to a second
system. Either of these systems may be electrical, mechanical or
acoustical.
Transmission.—Transmission in a system refers to the transmission of
power, voltage, current, force, velocity, torque, angular velocity, pressure
or volume current.
Transmission Loss {or Gain).—The transmission loss due to a system
joining a load having a given electrical, mechanical rectilineal, mechani-
cal rotational or acoustical impedance and a source having a given
electrical, mechanical rectilineal, mechanical rotational or acoustical
impedance and a given electromotive force, force, torque or pressure is
expressed by the logarithm of the ratio of the power delivered to the load
to the power delivered to the load under some reference condition. For
a loss the reference power is greater. For a gain the reference power is
smaller.
Decibel.—The abbreviation db is used for the decibel. The bel is the
fundamental division of a logarithmic scale expressing the ratio of two
amounts of power, the number of bels denoting such a ratio being the
logarithm to the base ten of this ratio. The decibel is one-tenth of a bel.
For example, with Pi and P2 designating two amounts of power and n
the number of decibels denoting their ratio
pK = 10 logio ^ , decibels
When the conditions are such that ratios of voltages or ratios of currents
(or analogous quantities such as forces or velocities, torques or angular
velocities, pressures or volume currents) are the square roots of the
corresponding power ratios, the number of decibels by which the corre-
sponding powers differ is expressed by the following formulas:
w = 20 logio T , decibels
n = 20 logio ^ ) decibels^2
where /1//2 and ^i/(?2 are the given current and voltage ratios respec-
tively.
CHAPTER II
ELEMENTS
2.1. Introduction
An element or circuit parameter in an electrical system defines a dis-
tinct activity in its part of the circuit. In an electrical system these
elements are resistance, inductance and capacitance. They are dis-
tinguished from the devices; resistor, inductor and capacitor. A resistor,
inductor and capacitor idealized to have only resistance, inductance and
capacitance is a circuit element. As indicated in the preceding chapter,
the study of mechanical and acoustical systems is facilitated by the
introduction of elements analogous to the elements of an electric circuit.
In this procedure, the first step is to develop the elements in these
vibrating systems. It is the purpose of this chapter to define and describe
electrical, mechanical rectilineal, mechanical rotational and acoustical
elements.'
2.2. Resistance
A. Electrical Resistance.—Electrical energy is changed into heat bythe passage of an electrical current through a resistance. Energy is lost
by the system when a charge q is driven through a resistance by a voltage
e. Resistance is the circuit element which causes dissipation.
Electrical resistance rs, in abohms, is defined as
ers^-. 2.1
t
where e = voltage across the resistance, in abvolts, and
/ = current through the resistance, in abamperes.
Equation 2.1 states that the electromotive force across an electrical
resistance is proportional to the electrical resistance and the current.
' See footnote 5, page 8.
12
RESISTANCE 13
B. Mechanical Rectilineal Resistance.—Mechanical rectilineal energy
is changed into heat by a rectilinear motion which is opposed by linear
resistance (friction). In a mechanical system dissipation is due to fric-
tion. Energy is lost by the system when a mechanical rectilineal resist-
ance is displaced a distance ^ by a force/jf
.
Mechanical rectilineal resistance (termed mechanical resistance) ru,
in mechanical ohms, is defined as
Vm = — 2.2u
where Jm — applied mechanical force, in dynes, and
u = velocity at the point of application of the force, in centi-
meters per second.
Equation 2.2 states that the driving force applied to a mechanical
rectilineal resistance is proportional to the mechanical rectilineal resist-
ance and the linear velocity.
C. Mechanical Rotational Resistance.—Mechanical rotational energy is
changed into heat by a rotational motion which is opposed by a rotational
resistance (rotational friction). Energy is lost by the system when a
mechanical rotational resistance is displaced by an angle (^ by a torque/^.
Mechanical rotational resistance (termed rotational resistance) vr, in
rotational ohms, is defined as
r^ = y 2.3
where Jr = applied torque, in dyne centimeters, and
d = angular velocity at the point of application about the axis,
in radians per second.
Equation 2.3 states that the driving torque applied to a mechanical
rotational resistance is proportional to the mechanical rotational resist-
ance and the angular velocity.
D. Acoustical Resistance.—In an acoustical system dissipation may be
due to the fluid resistance or radiation resistance. At this point the
former type of acoustical resistance will be considered. Acoustical
energy is changed into heat by the passage of a fluid through an acousti-
cal resistance. The resistance is due to viscosity. Energy is lost by the
system when a volume X is driven through an acoustical resistance by a
pressure p.
14 ELEMENTS
Acoustical resistance r^, in acoustical ohms, is defined as
PVA = ~ 2.4
where p — pressure, in dynes per square centimeter, and
U = volume current, in cubic centimeters per second.
Equation 2.4 states that the driving pressure applied to an acoustical
resistance is proportional to the acoustical resistance and the volume
current.
The transmission ofsound waves or direct currents of air through small
constrictions is primarily governed by acoustical resistance due to
viscosity. A tube of small diameter, a narrow slit, and metal, or cotton
or silk cloth are a few examples of systems which exhibit acoustical
resistance. There is also, in addition to the resistive component, a reac-
tive component. However, the ratio of the two components is a func-
tion of the dimensions. This is illustrated by the following equation for
the acoustic impedance ^ of a narrow slit.
1 Ijiw . 6pww
1H ^^'~Sld2A=^^+i-777 2.5
where /i = viscosity coefficient, 1.86 X 10"'* for air, density, in grams
per cubic centimeter,
d = thickness of the slit normal to the direction of flow, in
centimeters,
/ = width of the slit normal to the direction of flow, in centi-
meters,
w = length of the slit in the direction of flow, in centimeters,
w = 2wf, and
/ = frequency in cycles per second.
Any ratio of acoustical resistance to acoustical reactance can be
obtained by a suitable value o( d. Then the value of acoustical resistance
can be obtained by an appropriate value of w and /. The same expedient
may be employed in the case of any acoustical resistance in which the
resistance is due to viscosity.
'Olson, "Elements of Acoustical Engineering," D. Van Nostrand Co., NewYork, 1940.
INDUCTANCE, MASS, MOMENT OF INERTIA, INERTANCE 15
2.3. Inductance, Mass, Moment of Inertia, Inertance
A. Inductance.—Electromagnetic energy is associated with induct-
ance. Electromagnetic energy increases as the current in the inductance
increases. It decreases when the current decreases. It remains constant
when the current in the inductance is a constant. Inductance is the
electrical circuit element which opposes a change in current. Inductance
L, in abhenries, is defined as
^ die = L- 2.6
dt
where e = electromotive or driving force, in abvolts, and
di/di = rate of change of current, in abamperes per second.
Equation 2.6 states that the electromotive force across an inductance
is proportional to the inductance and the rate of change of current.
B. Mass.—Mechanical rectilineal inertial energy is associated with
mass in the mechanical rectilineal system. Mechanical rectilineal energy
increases as the linear velocity of a mass increases, that is, during linear
acceleration. It decreases when the velocity decreases. It remains con-
stant when the velocity is a constant. Mass is the mechanical element
which opposes a change of velocity. Mass m, in grams, is defined as
dufM = m^ 2.7
where du/dt = acceleration, in centimeters per second per second, and
Jm = driving force, in dynes.
Equation 2.7 states that the driving force applied to the mass is pro-
portional to the mass and the rate of change of linear velocity.
C. Moment of Inertia.—Mechanical rotational inertial energy is asso-
ciated with moment of inertia in the mechanical rotational system.
Mechanical rotational energy increases as the angular velocity of a
moment of inertia increases, that is, during angular acceleration. It
decreases when the angular velocity decreases. It remains a constant
when the angular velocity is a constant. Moment of inertia /, in gram
(centimeter)^, is given bydd
16 ELEMENTS
where ddjdt = angular acceleration, in radians per second per second,
and
Ju = torque, in dyne centimeters.
Equation 2.8 states that the driving torque applied to the moment of
inertia is proportional to the moment of inertia and the rate of change of
angular velocity.
D. Inertance.—Acoustical inertial energy is associated with inertance
in the acoustical system. Acoustical energy increases as the volumecurrent of an inertance increases. It decreases when the volume current
decreases. It remains constant when the volume current is a constant.
Inertance is the acoustical element that opposes a change in volumecurrent. Inertance M, in grams per (centimeter)*, is defined as
p = M—- 2.9dt
where M — inertance, in grams per (centimeter)*,
dU/dt = rate of change of volume current, in cubic centimeters per
second per second, and
p = driving pressure, in dynes per square centimeter.
Equation 2.9 states that the driving pressure applied to an inertance is
proportional to the inertance and the rate of change of volume current.
Inertance ^ may be expressed as
mM=—^ 2.10
where m = mass, in grams,
S = cross sectional area in square centimeters, over which the
driving pressure acts to drive the mass.
The inertance of a circular tube is
pi
where R = radius of the tube, in centimeters,
/ = effective length of the tube, that is, length plus end correc-
tion, in centimeters, and
p = density of the medium in the tube, in grams per cubic centi-
meter.
^ Olson, "Elements of Acoustical Engineering," D. Van Nostrand Co., NewYork, 1940.
CAPACITANCE AND COMPLIANCE 17
2.4. Electrical Capacitance, Rectilineal Compliance, Rotational Com-pliance, Acoustical Capacitance
A. Electrical Capacitance.—Electrostatic energy is associated with the
separation of positive and negative charges as in the case of the charges
on the two plates of an electrical capacitance. Electrostatic energy
increases as the charges of opposite polarity are separated. It is constant
and stored when the charges remain unchanged. It decreases as the
charges are brought together and the electrostatic energy released.
Electrical capacitance is the electrical circuit element which opposes a
change in voltage. Electrical capacitance Cg, in abfarads, is defined as
„ dei = Ce-: l.n
dt
Equation 2.12 may be written
= k f'idt= -~r 2.13
Le
where q = charge on electrical capacitance, in abcoulombs, and
e — electromotive force, m abvolts.
Equation 2.13 states that the charge on an electrical capacitance is
proportional to the electrical capacitance and the applied electromotive
force.
B. Rectilineal Compliance.—Mechanical rectilineal potential energy is
associated with the compression of a spring or compliant element. Me-chanical energy increases as the spring is compressed. It decreases as the
spring is allowed to expand. It is a constant, and is stored, when the
spring remains immovably compressed. Rectilineal compliance is the
mechanical element which opposes a change in the applied force. Rec-
tilineal compliance Cm (termed compliance) in centimeters per dyne, is
defined as
fM = jr 2.14
where x = displacement, in centimeters, and
Jm = applied force, in dynes
Equation 2.14 states that the linear displacement of a compliance is
proportional to the compliance and the applied force.
18 ELEMENTS
Stiffness is the reciprocal of compliance.
C. Rotational Compliance.—Mechanical rotational potential energy is
associated with the twisting of a spring or compliant element. Mechani-
cal energy increases as the spring is twisted. It decreases as the spring
is allowed to unwind. It is constant, and is stored when the spring
remains immovably twisted. Rotational compliance is the mechanical
element which opposes a change in the applied torque. Rotational
compliance Cu-, in radians per centimeter per dyne, is defined as
where <^ = angular displacement, in radians, and
Jr = applied torque, in dyne centimeters.
Equation 2.15 states that the rotational displacement of the rotational
compliance is proportional to the rotational compliance and the applied
force.
D. Acoustical Capacitance.—Acoustical potential energy is associated
with the compression of a fluid or gas. Acoustical energy increases as the
gas is compressed. It decreases as the gas is allowed to expand. It is
constant, and is stored when the gas remains immovably compressed.
Acoustical capacitance is the acoustic element which opposes a change in
the applied pressure. The pressure,^ in dynes per square centimeter, in
terms of the condensation, is
p = c^ps 2.16
where c — velocity, in centimeters per second,
p = density, in grams per cubic centimeter, and
s = condensation, defined in equation 2.17.
The condensation in a volume V due to a change in volume from V to
V - Vs =—^ 2.17
* Olson, "Elements of Acoustical Engineering," p. 9, D. Van Nostrand Co.,
New York, 1940.
REPRESENTATION OF ELEMENTS 19
The change in volume V — V' , in cubic centimeters, is equal to the
volume displacement, in cubic centimeters.
V - V = X 2.18
where X = volume displacement, in cubic centimeters.
From equations 2.16, 2.17, and 2.18 the pressure is
P = ~,X 2.19
Acoustical capacitance Ca is defined as
XP = ^r 2.20
where p = sound pressure in dynes per square centimeter, and
X = volume displacement, in cubic centimeters.
Equation 2.20 states the volume displacement in an acoustical capaci-
tance is proportional to the pressure and the acoustical capacitance.
From equations 2.19 and 2.20 the acoustical capacitance of a volume is
FCa = ^ 2.21
where F = volume, in cubic centimeters.
2.5. Representation of Electrical, Mechanical Rectilineal, Mechanical
Rotational and Acoustical Elements
Electrical, mechanical rectilineal, mechanical rotational and acoustical
elements have been defined in the preceding sections. Fig. 2.1 illustrates
schematically the four elements in each of the four systems.
The electrical elements, electrical resistance, inductance and electrical
capacitance are represented by the conventional symbols.
Mechanical rectilineal resistance is represented by sliding friction
which causes dissipation. Mechanical rotational resistance is represented
by a wheelwith a sliding friction brake which causes dissipation. Acousti-
cal resistance is represented by narrow slits which causes dissipation due
to viscosity when fluid is forced through the slits. These elements are
analogous to electrical resistance in the electrical system.
20 ELEMENTS
Inertia in the mechanical rectilineal system is represented by a mass.
Moment of inertia in the mechanical rotational system is represented by
a flywheel. Inertance in the acoustical system is represented as the fluid
contained in a tube in which all the particles move with the same phase
when actuated by a force due to pressure. These elements are analogous
to inductance in the electrical system.
—^AA/V-
-mm- -^
HhELECTRICAL ACOUSTICAL
RECTILINEAL ROTATIONAL
MECHANICAL
Fig. 2.1. Graphical representation of the three basic elements in electrical, mechanical
rectilineal, mechanical rotational and acoustical systems.
rE
REPRESENTATION OF ELEMENTS 21
O
<
22 ELEMENTS
TABLE 2.3
Electrical
REPRESENTATION OF ELEMENTS 23
TABLE 2.3—Continued
Mechanical Rotational
24 ELEMENTS
sions are mass, length, and time. These quantities are directly connected
to the mechanical rectilineal system. Other quantities in the mechanical
rectilineal system may be derived in terms of these dimensions. In terms
of analogies the dimensions in the electrical circuit corresponding to
length, mass and time in the mechanical rectilineal system are charge,
self-inductance, and time. The corresponding analogous dimensions in
the rotational mechanical system are angular displacement moment of
inertia, and time. The corresponding analogous dimensions in the acous-
tical system are volume displacement, inertance and time. The above
mentioned fundamental dimensions in each of the four systems are shownin tabular form in Table 2.1. Other quantities in each of the four systems
may be expressed in terms of the dimensions of Table 2.1.^ A few of the
most important quantities have been tabulated in Table 2.2. Tables 2.1
and 2.2 depict analogous quantities in each of the four systems. Further,
it shows that the four systems are dynamically analogous.
The dimensions given in Table 2.1 should not be confused with the
classical dimensions of electrical, mechanical and acoustical systems
given in Table 2.3. Table 2.3 uses mass M, length L and time T. In the
case of the electrical units dielectric and permeability constants are
assumed to be dimensionless.
^ The Tables 2.1, 1.1 and 2.3 deviate from the procedure outlined in footnote 5,
page 8, and list the standard modifiers for all four systems.
CHAPTER III
ELECTRICAL, MECHANICAL RECTILINEAL, MECHANICALROTATIONAL, AND ACOUSTICAL SYSTEMS
OF ONE DEGREE OF FREEDOM
3.1. Introduction
In the preceding sections the fundamental elements in each of the four
systems have been defined. From these definitions it is evident that
friction, mass, and compliance govern the movements of physical bodies
in the same manner that resistance, inductance and capacitance govern
the movement of electricity. In any dynamical system there are two
distinct problems; namely, the derivation of the differential equation
from the statement of the problem and the physical laws, and the solu-
tion of the differential equation. It is the purpose of this chapter to
establish and solve the differential equations for electrical, mechanical
rectilineal, mechanical rotational and acoustical systems of one degree
of freedom. These equations will show that the coefficients in the dif-
ferential equation of the electrical system are elements in the electrical
circuit. In the same way the coefficients in the differential equations of
the mechanical rectilineal, mechanical rotational and acoustical systems
may be looked upon as mechanical rectilineal, mechanical rotational or
acoustical elements. In other words, a consideration of the four sys-
tems of a single degree of freedom provides another means of establish-
ing the analogies between electrical, mechanical rectilineal, mechanical
rotational and acoustical systems.
3.2. Description of Systems of One Degree of Freedom
An electrical, mechanical rectilineal, mechanical rotational, and acous-
tical system of one degree of freedom is shown in Fig. 3.1. In one degree
of freedom the activity in every element of the system may be expressed
in terms of one variable. In the electrical system an electromotive force
e acts upon an inductance L, an electrical resistance Te and an electrical
25
26 SYSTEMS OF ONE DEGREE OF FREEDOM
capacitance Ce connected in series. In the mechanical rectilineal sys-
tem a driving force /m acts upon a particle of mass m fastened to a
spring or compliance Cm and sliding upon a plate with a frictional force
which is proportional to the velocity and designated as the mechanical
rectilineal resistance vm- In the mechanical rotational system a driving
torque fu acts upon a flywheel of moment of inertia / connected to a
spring or rotational compliance Cr and the periphery of the wheel sliding
against a brake with a frictional force which is proportional to the
velocity and designated as the mechanical rotational resistance vr. In
• TELECTRICAL
X
L-** m///>)»»//,
p M = T. C.
RECTILINEAL
I : Cr
FREQUENCY
ACOUSTICALROTATIONAL
MECHANICAL
Fig. 3.1. Electrical, mechanical rectilineal, mechanical rotational and acoustical systems
of one degree of freedom and the current, velocity, angular velocity and volume current
response characteristics.
the acoustical system an impinging sound wave of pressure p acts uponan inertance M and an acoustical resistance va comprising the air in the
tubular opening which is connected to the volume or acoustical capaci-
tance Ca- The acoustical resistance r^ is due to viscosity.
The principle of the conservation of energy forms one of the basic
theorems in most sciences. The principle of conservation of energy
states that the total store of energy of all forms remains a constant if
the system is isolated so that it neither receives nor gives out energy;
in case of transfer of energy the total gain or loss from the system is equal
to the loss or gain outside the system. In the electrical, mechanical
rectilineal, mechanical rotational, and acoustical systems energy will be
confined to three forms; namely, kinetic, potential and heat energy.
Kinetic energy of a system is that possessed by virtue of its velocity.
Potential energy of a system is that possessed by virtue of its configura-
tion or deformation. Heat is a transient form of energy. In the four
KINETIC ENERGY 27
systems; electrical, mechanical rectilineal, mechanical rotational, and
acoustical energy is transformed into heat in the dissipative part of
the system. The heat energy is carried away either by conduction or
radiation. The sum of the kinetic, potential, and heat energy during an
interval of time is, by the principle of conservation of energy, equal to
the energy delivered to the system during that interval.
3.3. Kinetic Energy
The kinetic energy Tke stored in the magnetic field of the electrical
circuit is
Tke = 2-^^ -^-l
where L = inductance, in abhenries, and
i — current through the inductance L, in abamperes.
The kinetic energy Tkm stored in the mass of the mechanical recti-
lineal system is
'iKM = 2Tkm = hmx"^ 3.2
where m = mass, in grams, and
X = velocity of the mass m, in centimeters per second.
The kinetic energy Tkr stored in the moment of inertia of the mechani-
cal rotational system is
Tkr = 2-^s!>" 3.3
where / = moment of inertia, in gram (centimeter)^ and
4> = angular velocity of /, in radians per second.
The kinetic energy Tka stored in the inertance of the acoustical
system is
Tka = iMX'^ 3.4
where M = m/S^, the inertance, in grams per (centimeter)*,
m = mass of air in the opening, in grams,
S — cross-sectional area of the opening, in square centimeters,
X — Sx = volume current, in cubic centimeters per second,
X = velocity of the air particles in the opening, in centimeters
per second.
It is assumed that all the air particles in the opening move with the
same phase.
28 SYSTEMS OF ONE DEGREE OF FREEDOM
3.4. Potential Energy
The potential energy Vpe stored in the electrical capacitance of the
electrical circuit is
where Ce = capacitance, in abfarads, and
q = charge on the capacitance, in abcoulombs.
The potential energy VpM stored in the compliance or spring of the
mechanical rectilineal system is
1 x^
I Lm
where Cm = lA = compliance of the spring, in centimeters per dyne,
s — stiffness of the spring, in dynes per centimeter, and
X = displacement, in centimeters.
The potential energy Fpn stored in the rotational compliance or spring
of the mechanical rotational system is
FpJ^ = -^ 3.72Cu
where Cr = rotational compliance of the spring, in radians per dyne per
centimeter, and
<l>= angular displacement, in radians.
The potential energy Fpa stored in the acoustical capacitance of the
acoustical system is
1 v2v.. =
-, ^ 3.8
where X = volume displacement, in cubic centimeters,
Ca = V/pc^= acoustical capacitance, in (centimeters)® per dyne,
V = volume of the cavity, in cubic centimeters,
p = density of air, in grams per cubic centimeter, and
c = velocity of sound, in centimeters per second.
DISSIPATION 29
The energies stored in the systems is the sum of the kinetic and poten-
tial energy. The total energy stored in the four systems may be written
We = Tke + VpE = hLP + \^ 3.91 Le
Wm = Tkm + ^PM = Imx^ + ;: 7^ 3.102 Cm
Wr = Tkr + VpR = i/02 + 1 ^ 3.11
/?^A = T^A + VpA = \MX^ + \ ^ 3.12
where We, Wm, Wr, and Wa are the total energies stored in electrical,
mechanical rectilineal, mechanical rotational, and acoustical systems.
The rate of change of energy with respect to time in the four systems
may be written
^^ = Z/- + -^ = L-- + -^ 3 13dt dt Ce Ce
dWu XX—-— = mxx + —
—
3.14dt Cm
dWp ^. .. <j)^-/ = I<t>^ + ~^ 3.15dt Cr
dWA v.y- XX~-^ = MXX + -^ 3.16at La
3.5. Dissipation
The rate at which electromagnetic energy De is converted into heat is
Be = rsi^ 3.17
where Ta = electrical resistance, in abohms, and
/ = current, in abamperes.
Assume that the frictional force.Jm upon the mass m as it slides back
and forth is proportional to the velocity as follows:
Jm = rMX 3.18
30 SYSTEMS OF ONE DEGREE OF FREEDOM
where tm = mechanical resistance, in mechanical ohms, and
X = velocity, in centimeters per second.
The rate at which mechanical rectilineal energy Dm is converted into
heat is
Dm = /mx = tmx^ 3.19
Assume that the frictional torque /r upon the flywheel / as the pe-
riphery of the wheel slides against the brake is proportional to the veloc-
ity as follows:
/b = rR4> 3.20
where vr = mechanical rotational resistance, in rotational ohms, and
4> = angular velocity, in radians per second.
The rate at which mechanical rotational energy Dr is converted into
heat is
DR=fR4> = rR4? 3.21
The acoustical energy is converted into heat by the dissipation due to
viscosity as the fluid is forced through the narrow slits. The rate at
which acoustical energy Da is converted into heat is
Da = r^X^ 3.22
where ta = acoustical resistance, in acoustical ohms, and
X = volume current in cubic centimeters per second.
3.6. Equations of Motion
The power delivered to a system must be equal to the rate of kinetic
energy storage plus the rate of potential energy storage plus the power
loss due to dissipation. The rate at which work is done or power de-
livered to the electrical system by the applied electromotive force is
^Ee-""* = eq. The rate at which work is done or power delivered to the
mechanical rectilineal system by the applied mechanical force is
xpMf?''* = JmX. The rate at which work is done or power delivered to
the mechanical rotational system by the applied mechanical torque is
<j)FRe^"^ = Jri^i. The rate at which work is done or power delivered to
the acoustical system by the applied sound pressure is XPe-""' = pX.The rate of decrease of energy {Tk + Vp) of the system plus the rate
at which work is done on the system or power delivered to the system
EQUATIONS OF MOTION 31
by the external forces must equal the rate of dissipation of energy.
Writing this sentence mathematically yields the equations of motion for
the four systems.
Electrical
Lqq + VE^ + ff = £*'"'? 3.23Le
Lq + rEq + jr = Ee^"* 3.24Le
Mechanical Rectilineal
mxx + ru>i^ + 7^ = Fm^^^^x 3.25
Xmx -\- Tmx + ~pr~
~ Piie'"^ 3.26Cm
Mechanical Rotational
m + ra^^ + ^ = Fae''-"^ 3.27
Iii> + rR4> + ^ = Fee'"'' 3.28
Acoustical
XXMXX + taX^ + -tt = P^'^^'X 3.29Ca
MX + r^X + ^ = Pe^'"' 3.30
The steady state solutions of the four differential equations 3.24, 3.26,
3.28 and 3.30 are
Electrical
Ft''"' e
q = I = :— = — 3.31
te -\-joiL - —
-
'^E
Mechanical Rectilineal
•
^^^'"' Im „..X = r- = — 3.32
JO^ Zmrm +jccm — -—-
Cm
32 SYSTEMS OF ONE DEGREE OF FREEDOM
Mechanical Rotational
^ = — :- = ^-3.33
J<^ zr
Acoustical
ra+M ^
X = r- = — 3.34
r.4 + jwM — —
The vector electrical impedance is
ze = rE + joiL - — 3.35
The vector mechanical rectilineal impedance is
_7uzm = rR-\- jwm — ~~- 3.36
The vector mechanical rotational impedance is
zr = rR-\- jo>I — — 3.37
The vector acoustical impedance is
ZA = r^+icoTkf- 7^ 3.38Ca
3.7. Resonant Frequency
For a certain value of L and Ce, m and Cm, -^ and Cr, and Af and Cathere will be a certain frequency at which the imaginary component of
the impedance is zero. This frequency is called the resonant frequency.
At this frequency the ratio of the current to the applied voltage or the
ratio of the velocity to the applied force or the ratio of the angular
velocity to the applied torque or the ratio of the volume current to the
applied pressure is a maximum. At the resonant frequency the current
and voltage, the velocity and force, the angular velocity and torque,
and the volume current and pressure are in phase.
KIRCHHOFF'S LAW AND D'ALEMBERT'S PRINCIPLE 33
The resonant frequency/r in the four systems is
Electrical
It = 7== 3.39iWLCe
Mechanical Rectilineal
Mechanical Rotational
Acoustical
Jr = i= 3.4027rV mCM
iWYcr
fr = i= 3.42iWmca
3.8. Kirchhofif's Law and D'Alembert's Principle '
KirchhofF's electromotive force law plays the same role in setting up
the electrical equations as D'Alembert's principle does in setting up
mechanical and acoustical equations. It is the purpose of this section
to obtain the differential equations of electrical, mechanical rectilineal,
mechanical rotational and acoustical systems employing Kirchhoff's
law and D'Alembert's principle.
KirchhofF's law is as follows: The algebraic sum of the electromotive
forces around a closed circuit is zero. The differential equations for
electric circuits with lumped elements may be set up employing Kirch-
hofF's law. The electromotive forces due to the elements in an electric
circuit are
Electromotive force of self-inductance = —L—-= —L—r^ 3.43
Electromotive force of electrical resistance = — rg/ = — r^ — 3.44dt
Electromotive force of electrical capacitance =—--;- 3.45Ce
In addition to the above electromotive forces are the electromotive
forces applied externally.
^ D'Alembert's principle as used here may be said to be a modified form of
Newton's second law.
34 SYSTEMS OF ONE DEGREE OF FREEDOM
The above law may be used to derive the differential equation for the
electrical circuit of Fig. 3.1. From Kirchhoff's law the algebraic sum of
the electromotive forces around the circuit is zero. The equation maybe written
L~ + TEt + -^ = Et'''* 3A6dt Ce
where e = £e^"' = the external applied electromotive force.
Equation 3.46 may be written
and is the same as equation 3.24.
The differential equations for mechanical systems may be set upemploying D'Alembert's principle; namely, the algebraic sum of the
forces applied to a body is zero.
The mechanical forces due to the elements in a mechanical rectilineal
system are
d'^xMechanomotive force of mass reaction = —ni~-^ 3.48
dr
Mechanomotive force of mechanical rectilineal resistance =dx „ ,„-ru— 3.49dt
XMechanomotive force of mechanical compliance = — —
—
3.50Cm
In addition to the above mechanomotive forces are the mechano-
motive forces applied externally.
The above principle may be used to derive the differential equation of
the mechanical rectilineal system of Fig. 3.1. From D'Alembert's
principle the algebraic sum of the forces applied to a body is zero. Theequation may be written
where _/Af = F^e-'"' = external applied mechanical force
^^ + rM^^-^ = Fm^''"' 3.51at at C/i:
KIRCHHOFF'S LAW AND D'ALEMBERT'S PRINCIPLE 35
Equation 3.51 is the same as equation 3.26.
D'Alembert's principle may be applied to the mechanical rotational
system. The rotational mechanical forces due to the elements in a
mechanical rotational system are
Rotatomotive force of moment of inertia reaction = — /—rs 3.52dr
Rotatomotive force of mechanical rotational resistance =dcf)
-ru — 3.53at
<t>
Rotatomotive force of rotational compliance = — —
-
3.54Cr
In addition to the above rotatomotive forces are the rotatomotive
forces applied externally.
Applying D'Alembert's principle the equation for the rotational sys-
tem of Fig. 3.1 may be written
<*2 ' '"' dt ' CrZ^ + rij-f + ~ = i^ije^'"' 3.55
where /ij = pRe^"' = external applied torque.
Equation 3.55 is the same as equation 3.28.
D'Alembert's principle may be applied to the acoustical system.
The acoustical pressures due to the elements in an acoustical system are
d^XAcoustomotive force of inertive reaction = — Af —tit 3.56
dr
dxAcoustomotive force of acoustical resistance = — r^ — 3.57
d(
Acoustomotive force of acoustical capacitance = — — 3.58Ca
In addition to the above acoustomotive forces are the acoustomo-
tive forces applied externally.
36 SYSTEMS OF ONE DEGREE OF FREEDOM
Applying D'Alembert's principle, the equation for the acoustical
system of Fig. 3.1 may be written
M^+r^^+^ = Pe-' 3.59dr dt Ca
where p = Pe^"' = external applied pressure.
Equation 3.59 is the same as equation 3.30.
Equations 3.43 to 3.59, inclusively, further illustrate the analogies
between electrical, mechanical rectilineal, mechanical rotational, and
acoustical systems.
CHAPTER IV
ELECTRICAL, MECHANICAL RECTILINEAL, MECHANICALROTATIONAL AND ACOUSTICAL SYSTEMS OF TWO
AND THREE DEGREES OF FREEDOM
4.1. Introduction
The analogies between the four types of vibrating systems of one
degree of freedom have been considered in the preceding chapter. It is
the purpose of this section to extend these analogies to systems of two
and three degrees of freedom. In this chapter the differential equations
for the four systems will be obtained from the expressions for the kinetic
and potential energies, the dissipation and the application of Lagrange's
equations.
4.2. Two Degrees of Freedom
The first consideration will be the systems shown in Fig. 4.1. In the
electrical system an electromotive force acts upon an electrical capaci-
-HiELECTRICAL
ACOUSTICAL
RECTILINEAL
ROTATIONAL
MECHANICAL
FREQUENCY
Fig. 4.1. Electrical, mechanical rectilineal, mechanical rotational and acoustical systemsof two degrees of freedom and the input current, velocity, angular velocity and volumecurrent response characteristics.
tance Ce shunted by an inductance L and an electrical resistance re in
series. In the mechanical rectilineal system a driving force acts upon a
37
38 SYSTEMS OF 2 AND 3 DEGREES OF FREEDOM
spring or compliance Cj^i connected to a mass m sliding upon a plate with
a frictional force which is proportional to the velocity and designated as
the mechanical rectilineal resistance r^- In the mechanical rotational
system a driving torque acts upon a spring or rotational compliance Crconnected to a flywheel of moment of inertia / and with the periphery of
the wheel sliding against a brake with a frictional force which is propor-
tional to the velocity and designated as the mechanical rotational resist-
ance tr. In the acoustical system a driving pressure p acts upon a
volume or acoustical capacitance Ca connected to a tubular opening
communicating with free space. The mass of fluid in the opening is the
inertance M and the fluid resistance produced by the slits is the acousti-
cal resistance r^-
4.3. Kinetic Energy
The kinetic energy Tke stored in the magnetic field of the electrical
circuit is
Tke = i^?3' 4.1
where L = inductance, in abhenries, and
93 = h = current, in branch 3, in abamperes.
The kinetic energy Tkm stored in the mass of the mechanical rectilineal
system is
Tkm = lmx:i^ 4.2
where m = mass, in grams, and
x^ = velocity of the mass m, in centimeters per second.
The kinetic energy Tkr stored in the moment of inertia of the mechani-
cal rotational system is
Tkr = 2l4>3 4.3
where / = moment of inertia, in gram (centimeter)^ and
<ji3 = angular velocity of /, in radians per second.
The kinetic energy Tka stored in the inertance of the acoustical sys-
tem is
Tka = hMX^ 4.4
where M = inertance, in grams per (centimeter)* and
X^ = volume current, in cubic centimeters per second.
DISSIPATION 39
4.4. Potential Energy
The potential energy VpE stored in the electric field of the electrical
circuit is
J^PE =~^f~
4.5
where Ce = capacitance, in abfarads, and
92 = charge on the electrical capacitance, in abcoulombs.
The potential energy Vp^ij stored in the compliance or spring of the
mechanical rectilineal system is
VpM =T TT 4.6
where Cm = compliance of the spring, in centimeters per dyne, and
X2 — displacement, in centimeters.
The potential energy VpR stored in the rotational compliance or spring
of the mechanical rotational system is
where Cu — rotational compliance of the spring, in radians per dyne per
centimeter, and
<i>2= angular displacement, in radians.
The potential energy Fpa stored in the acoustical capacitance of the
acoustical system is
where Ca = acoustical capacitance, in (centimeter)^ per dyne, and
X2 = volume displacement, in cubic centimeters.
4.5. Dissipation
The rate at which electromagnetic energy De is converted into heat is
De = rsh'^ = TEqz 4.9
where rg = electrical resistance, in ohms, and
4 = qz — current, in abamperes.
40 SYSTEMS OF 2 AND 3 DEGREES OF FREEDOM
The rate at which mechanical rectihneal energy Dm is converted into
heat is
Dm = VMXz^ 4.10
where Vm = mechanical rectilineal resistance, in mechanical ohms, and
X2, = velocity, in centimeters per second.
The rate at which mechanical rotational energy Dr is converted into
heat is
Dr = vr^^ 4.11
where tr = mechanical rotational resistance, in rotational ohms, and
cj>z = angular velocity, in radians per second.
The rate at which acoustical energy Da is converted into heat is
Da = rAXs^ 4.12
where r.i = acoustical resistance, in acoustical ohms, and
Xs = volume current, in cubic centimeters per second.
4.6. Equations of Motion
Lagrange's equations for the four systems are as follows:
Electrical
d /'dT\ d(T- V) 1 dD
dt \dqn/ aqn 2 dqn
where n = number independent coordinates.
Mechanical Rectilineal
a /dT\ _ d{T-V) ^IdD^^^^^_j^
dt \dXn/ dXn 2 dXn
Mechanical Rotational
d<t>n 2 dcj),
Acoustical
'<^-^''^^=A.
d / dT\ d{T - V) I dD
dt \dXn/ dXn 2 dXn
THE MECHANICAL RECTILINEAL SYSTEM 41
4.7. The Electrical System
Applying Lagrange's equation 4.13,
e = Lqz-[- rsqz 4.17
' = IThe electromotive force applied to the inductance and electrical re-
sistance in series is given by equation 4.17. The electromotive force
applied to the electrical capacitance in terms of the displacement is
given by equation 4.18.
The relation for the currents in Fig. 4.1 is
h = H + k 4.19
Equation 4.19 may be written
9i = ?2 + ?3 or 4.20
5'i = ?2 + ?3 4.21
4.8. The Mechanical Rectilineal System
Applying Lagrange's equation 4.14,
Jm = mx^ + ruxz 4.22
hi = / 4.23
The force applied to the mass and mechanical rectilineal resistance is
given by equation 4.22. The force applied to the spring in terms of the
displacement is given by equation 4.23.
The linear displacement, at/jf, in the mechanical rectilineal system
of Fig. 4.1 is the sum of the displacement of the mass m and the dis-
placement of the compliance Cm-
s^i = ^2 + Xo, 4.24
Differentiating equation 4.24 with respect to the time the velocities
are
•*i = •*2 + -^s 4.25
42 SYSTEMS OF 2 AND 3 DEGREES OF FREEDOM
4.9. The Mechanical Rotational System
Applying Lagrange's equation 4.15,
Jb = /03 + rij^g 4.26
/. = ^ 4.27
The torque applied to the flywheel and mechanical rotational resist-
ance is given by equation 4.26. The torque applied to the spring is
given by equation 4.27.
The angular displacement, at/^j, in the mechanical rectilineal system
of Fig. 4.1 is the sum of the angular displacement of the flywheel / and
the angular displacement of the rotational compliance Cr.
<Pi= 4>2 + <t>3 4.28
Differentiating equation 4.28 with respect to the time the angular
velocities are
4>i = 4>2 + ^3 4.29
4.10. The Acoustical System
Applying Lagrange's equation 4.16,
p = MXa + vaXz 4.30
The pressure applied to the inertance and acoustical resistance is
given by equation 4.30. The pressure applied to the acoustical capaci-
tance in terms of the volume displacement is given by equation 4.31.
The volume displacement, at p, in the acoustical system, Fig. 4.1, is
the sum of the volume displacement of the inertanceM and the volume
displacement of the acoustic capacitance Ca-
Xi = X2 + X3 4.32
The volume displacement Xi is the volume displacement of the vibrat-
ing piston. The vibrating piston is not a part of the acoustical system.
It is merely the sound pressure source which produces the sound pres-
sure p.
COMPARISON OF THE FOUR SYSTEMS 43
Differentiating equation 4.32 with respect to the time the volume
currents are
Zi = ^2 + Xa 4.33
4.11. Comparison of the Four Systems
A comparison of the coefficients of equations 4.1 to 4.33, inclusive,
shows again that resistance, inductance, and capacitance are analogous
to mechanical rectilineal resistance, mass, and compliance in the mechan-
ical rectilineal system, to mechanical rotational resistance, moment of
inertia and rotational compliance in the mechanical rotational system,
and to acoustical resistance, inertance and acoustic capacitance in the
acoustical system. A comparison of equations 4.19, 4.25, 4.29 and 4.33
shows that currents in the electrical system are analogous to velocities
in the mechanical rectilineal system, to angular velocities in the mechan-
ical rotational system, and to volume currents in the acoustical system.
The current i^ through the inductance L and electrical resistance rg.
Fig. 4.1, is given by^
4.34«3 =
The total current /'i is given by
n = t±EL 4.35
(re + jwL) -T—rJuLm
The current 12 through the electrical capacitance Cg is
4 = h — h = ejwCE 4.36
The linear velocity *3 of the mass m and mechanical rectilineal resist-
ance, vm. Fig. 4.1, is given by r
JM^3
The linear velocity X\ ntfM is given by
Jm ( tm + jo^m + -T-— )
4.37
XijcoCAf/
4_3g
JwLm
44 SYSTEMS OF 2 AND 3 DEGREES OF FREEDOM
The velocity X2, the difference in hnear velocity between the two ends
of the spring Cm, is given by
*2 = *! — 'fs = /mJ<^Cm 4.39
The angular velocity <j!)3 of the moment of inertia / and mechanical
rotational resistance rg. Fig. 4.1, is given by
k =f". , 4.40
The total angular velocity ^i at/i? is given by
/R\rB + jojl + -7-^)
^1 = —^ i^:^ 4.41
(rs + jwl) ^—
-
The angular velocity 4>2, the difference in angular velocity between the
two ends of the spring Cn, is given by
h = ^1 - 4>3 = fRJ(^CR 4A2
The volume current X3 through the inertance M and the acoustic
resistance ta, Fig. 4.1, is given by
^3 = / ,. 4.43
The total volume current Xi at ^ is given by
pirA+ juM + ^— )
X, = -^ Zff£iZ 4.44
(rA + jwM) -T-z^rjuLa
The volume current X2, the difference between the volume currents at
the input and output to the acoustical capacitance, is
X2 = Xi - X3 = pjccCa 4.45
CAPACITIVE COUPLED SYSTEM 45
4.12. Electrical Inductive and Capacitive Coupled Systems of TwoDegrees of Freedom and the Mechanical Rectilineal, Mechani-cal Rotational and Acoustical Analogies
It is the purpose of this section to show two additional electrical
arrangements of two degrees of freedom and the mechanical rectilineal,^
mechanical rotation and acoustical analogies.
The electrical impedances z^i, 2^2 and z^s in terms of the elements of
Fig. 4.2 are as follows:
jwCiEl
ZE2 = 1
JuCe2
ZE3 = rE2 + j<^L2 +JwCe3
4.46
4.47
4.48
Cr,
|-/35^ |-vwv—p-/55ir^
I 1
ELECTRICAL
P M,=
ACOUSTICAL
ROTATIONAL
MECHANICAL
Fig. 4.2. A capacitive coupled electrical system of two degrees of freedom and themechanical rectihneal, mechanical rotational and acoustical analogies. The graphdepicts the output response frequency characteristic.
1 For an explanation of the shunt mechanical rectilineal and mechanical rota-tional systems of Figs. 4.2 and 4.3, see pages 53, 54, 55 and 56 and Fig. 5.1 ofChapter V.
46 SYSTEMS OF 2 AND 3 DEGREES OF FREEDOM
The mechanical rectilineal impedances zmi, zm2 and zm3 in terms of
the elements of Fig. 4.2 are as follows:
Zmi = rui + jwmi + ^—— 4.49
2M2 = ^-pr- 4.50JwLm2
2m3 = rM2 + jwm2 + T-p,— 4.51JwLmz
The mechanical rotational impedances zm, Zb,2 and Zrz in terms of the
elements of Fig. 4.2 are as follows:
ZiJi = rui +ico/i + . 4.52
Zij2 = —TT- 4.53
Zijs = rK2 + y'0/2 + -T-zT- 4.54
The acoustical impedances z^i, 2^2) and 2^3 in terms of the elements of
Fig. 4.2 are as follows:
ZAi = rAi +icoMi + r-—
-
4.55jwLai
ZA2 = -T-z;- 4.56JuLa2
za3 = rA2 + iwMa + ^—
-
4.577C0C43
The system in Fig. 4.3 is the same as that of Fig. 4.2 save that the
shunt electrical capacitance, compliance, rotational compliance and
acoustical capacitance, Ce2, Cm2i Cr2 and Ca2i are replaced by the shunt
inductance, mass, moment of inertia and inertance L2, 1712, I2 and M2.
The shunt electrical, mechanical rectilineal, mechanical rotational and
acoustical shunt impedances are
ze2 = j^L2 4.58
2m2 = ji^m2 4.59
ZiS2 = j^l2 4.60
2^2 = jwM2 4.61
INDUCTIVE COUPLED SYSTEM
The current in the branch z^i is
e{ZE2 + Zes)^1
==1 ]ZeiZE2 + ZeiZE3 + Ze2ZE3
The velocity of the mass mi is
fM(ZM2 + Zms)Xi =
ZmiZM2 + ZmiZms + ZM2ZM3
47
4.62
4.63
ROTATIONAL
MECHANICAL
Fig. 4.3. An inductive coupled electrical system of two degrees of freedom and the
mechanical rectihneal, mechanical rotational and acoustical analogies. The graph
depicts the output response frequency characteristic.
The angular velocity of the moment of inertia /i is
/r(ZR2 + Zrs)<^1 = -
ZriZR2 + ZRiZRa + Zs2Zb3
The volume current of the inertance Mi is
piZA2 + Zas)Xi =
ZaiZA2 + ZaiZA3 + 2^22^3
4.64
4.65
48 SYSTEMS OF 2 AND 3 DEGREES OF FREEDOM
The current in the branch z^s is
h =]
; 4.66ZeiZE2 -f- Z£:i2£3 + Ze2Z£3
The velocity of the mass m^ is
JmZM2 . ,tX3 = -^ 4.67
The angular velocity of the moment of inertia lo, is
^3 = ^^^M^.^ 4.68Z/eiZB2 + ZiJl2iJ3 + ZiJ2ZiJ3
The volume current of the inertance Ms is
X, = ^M^^ 4.69ZaiZA2 + 2.412^3 + ZA2ZA3
The response frequency characteristic of the system is shown in
Figs. 4.2 and 4.3.
4.13. Electrical, Mechanical Rectilineal, Mechanical Rotational andAcoustical Systems of Three Degrees of Freedom
Systems of three degrees of freedom are shown in Fig. 4.4. Following
the procedures outlined in the preceding sections it can be shown that
RECTILINEAL
ELECTRICAL
_Cri 4i^i Cb2 ^2
y Ca, O Ca2 iJfTo *^l '"a= '^2
ROTATIONAL
ACOUSTICALMECHANICAL
Fig. 4.4. Electrical, mechanical rectilineal, mechanical rotational and acoustical systems
of three degrees of freedom.
L,\, L2, Cei, Ce2 and rg in the electrical system are equivalent to ;«i, m2,
Cm\, Cm2 and tm in the mechanical rectilineal system, to /i, I2, Cri, Cb2
SYSTEMS OF 3 DEGREES OF FREEDOM 49
and vr in the mechanical rotational system and to Mi, M2, Cai, Ca2
and rA in the acoustical system. These equations also show that /q,
h, h, h and i^ in the electrical system are equivalent to Xq, Xi, X2, X3
and Xi in the mechanical rectilineal system, to 4>o, 4>i, 4>2, (h and 04 in
the mechanical rotational system and to Xq, Xi, X2, X^ and to X4 in
the acoustical system.
The current to, the linear velocity Xq, the angular velocity ^ and the
volume current Xq are given by
e[(ZEi + Ze2){zB3 + ZBi) + ZE3ZE4] . -^to = 4.70
xo = 4.71
/rKzri + 2722) (Zij3 + ZR4) + ZR3ZR4]00 = 7; *-/-i
nA
where zei =. ^ zjji = . 4.74 4.75
2iS2 =iw/i 4.76 4.77
ZR3 = T^ 4.78 4.79
ZR2 = rR ^jwh 4.80 4.81
ZAx = ——
-
4.82 4.83
ZA2 = >Mi 4.84 4.85
zmz = ^-^ 243 = ^-p7- 4.86 4.87
ZMi = ru +jcom2 za4 = rA +J01M2 4.88 4.89
50 SYSTEMS OF 2 AND 3 DEGREES OF FREEDOM
He = ZElZEZ'ZEi + ZeiZe2{zE3 + ZEi) 4.90
Hm = ZmiZm3ZM4: + ZmiZM2(.ZM3 + ZMi) 4.91
Hr = ZmZR3ZR4. + ZRiZE2izR3 + ZRi) 4.92
Ha = ZA1ZA3ZA4: + ZaiZa2(.ZA3 + ZAi) 4.93
The current ii, the linear velocity Xi, the angular 4>i and the volume
current Xi are given by
e[ZE2iZE3 + ZEi) + ZE3ZE4:]
H = 4.94flE
/M[ZM2iZM3 + ZAfi) + ZAf^ZMi] . _.^1 = 4.95
MzR2(Zr3 + ZRi) + ZR^ZRil01
=t; 4.y6
Xi = -^--^-° —4.97
tlA
The current i2, the linear velocity X2, the angular velocity ^2) and the
volume current X2 are given by
eZEl{ZE3 + 2^4) . „ot2 = 7} 4.98
Me
fMZMl{ZM3 + ZM'd . r^r^X2 = 4.99
., fRZR\{zR3 + Zr^02 = Yt 4.1UU
Hr
^^^ pZA.{ZA3jrZA.)^_^^j
Ha
SYSTEMS OF 3 DEGREES OF FREEDOM 51
The current /'s, the Hnear velocity x^, the angular velocity ^3 and the
volume current X^ are given by
iz = -ij^ 4.102He
X3 = 77 4.103
4>z = —7}— 4.104
Xs = ^^^ 4.105
The current 4, the linear velocity X4, the angular velocity 04 and the
volume current X^ are given by
eZElZE3. ,„,
li = -- -- 4.106riE
JmZmiZmz . i„_Xi = — 4.107
J~iR
X, = ^^^ 4.109
The equations in this section show that the equations for the electri-
cal, mechanical rectilineal, mechanical rotational and acoustical systems
are similar and analogous.
CHAPTER V
CORRECTIVE NETWORKS
5.1. Introduction
A corrective network is a structure which has a transmission char-
acteristic that is more or less gradual in slope. Such a characteristic is
obtained when an inductance, electrical capacitance or the combination
of both is shunted across a line.^ Another type of corrective network is
an inductance, electrical capacitance or combination of both connected
in series with a line. Resistance may be introduced as a second or third
element in either shunt or series corrective networks. Various types of
resistance networks may be used as attenuators or for matching dissimi-
lar impedances. It is the purpose of this chapter to illustrate further
analogies between electrical, mechanical rectilineal, mechanical rota-
tional, and acoustical systems having similar transmission characteristics.
5.2. Two Electrical, Mechanical Rectilineal, Mechanical Rotational or
Acoustical Impedances in Parallel
Two electrical impedances zei and 2^2 are shown in parallel in Fig. 5.1.
The electrical impedance Zet of the two electrical impedances in parallel
is given by
Z_EiZE2Zet
Zei + 2^:2
5.1
The electromotive force e across zgr is also the electromotive force
across zei and Ze2-
The total current It is the vector sum of the currents ii and «2 as
follows
:
h = h + h 5.2
' The term "line" is used in this chapter to designate an electrical networkwhich, prior to the introduction of the corrective network, consisted of a gen-
erator in series with two electrical impedances, termed the input and output
electrical impedances.
52
IMPEDANCES IN PARALLEL 53
If 2;£2 is made infinite the current 22 in this branch is zero and the total
current flows in z^i, that is, It = i\- In the same way if z^i is madeinfinite the current /i in this branch is zero and the total current flows
in ze2i that is, ir = 12-
The mechanical rectilineal system, Fig. 5.1, consists of a system of
rigid massless levers and links with frictionless bearings at the connecting
54 CORRECTIVE NETWORKS
The linear displacement xt at 1 (for small amplitudes) is equal to the
vector sum of the displacements Xi and X2 of points 3 and 4, respectively.
Xt = xx -\- X2 5.3
Differentiating equation 5.3,
Xt = Xi -\- X2 5.4
That is, the linear velocity xt at / is equal to the vector sum of the linear
velocities Xi and X2 at points 4 and 3, respectively. Equation 5.4 is
analogous to equation 5.2 for the electrical system.
Since the same force /if exists at points 3 and 4 as the driving point
and further since the velocity at 1 is the vector sum of the velocities at
points 3 and 4, the mechanical impedances zm2 and zm\ at points 3 and 4,
respectively, must be in parallel. That is, the mechanical impedance
zmt at point 1 \s „ „
ZMT = , 5.5Zm\ + Zm2
If 23/2 is made infinite there can be no motion at point 3 and therefore
the system behaves the same as if zj/i were connected to the point /.
In the same way if Z3/1 is made infinite there is no motion at point 4 and
the system behaves the same as if Zm2 were connected directly to the
point 1.
The mechanical rotational system. Fig. 5.1, consists of a system of
planetary 2 gears. The diameters of gears i and 2 are equal. The inside
diameter of the large gear 3 is three times the gears 1 and 2. The outside
diameter of the gear 4 is three times that of gear 5. The diameter of
gear 6 is four times that of gear 7. Gear 2 is free to rotate on its shaft.
^ Practically all rotational systems which connect two mechanical rotational
systems in parallel are of the epicyclic train type. In this book the planetary
system is used to depict and illustrate a rotational system which connects me-chanical rotational impedances in parallel. There are other examples of epicyclic
trains which connect mechanical rotational impedances in parallel, as for ex-
ample, the differential used in automobiles and tractors, shown schematically in
Fig. SAA. The gear 2 is keyed to the shaft /. The gear 2 drives the ring gear 3.
The gears 4 and 9 rotate on bearings in the ring gear 3. The ring gear 3 rotates
freely on the shaft 8. Gears 4 and 9 drive the gears 5 and 6. Gears 5 and 6 are
keyed to the shafts 7 and <?, respectively. The diameter of gear 2 is one-half of
the diameter of ring gear 3. With these specifications the differential of Fig.
SAA performs the same function as the planetary system of Fig. 5.1 with the
same relations existing between//;, </)r, </>!, ^2, Z/jt, Zki and Zr2 in both illustra-
tions.
IMPEDANCES IN PARALLEL 55
The large gear 3-4 is free to rotate with its axis coincident with gear /.
The remainder of the gears are keyed to the respective shafts. Under
these conditions if gear 7 is held stationary the angular displacement of
gear 5 is the same as the driving gear 1. Or if 5 is held stationary the
angular displacement of gear 7 is the same as the driving gear 1. In all
the considerations which follow it is assumed that the ratios for the
\\\\\<\w£m^^
x
Zri2I
J2&
^ ti
END VIEW PLAN VIEW
Fig. i.\A. Differential gear train which connects two mechanical rotational impedances
in parallel. This system accomplishes the same results as the planetary system of
Fig. S.l.
various gears are as outlined above. In addition it is assumed that all
the gears are massless and that all the bearings are frictionless.
Since the gears are massless, the torque at gears 7 and 5 for driving the
rotational impedances zrx and 2ij2 is the same as the applied torque.
This is analogous to the same electromotive force across the impedances
Ze\ and Ze2 in the electrical circuit.
The angular displacement 4>t at gear 1 is equal to the vector sum of the
angular displacement at i>\ and<i>-2,
of the gears 7 and 5, respectively.
•Ar = 01 + </>2 5.6
Differentiating equation 5.6,
<i>\ + 'J>2 5.7
That is, the angular velocity ^r at 1 is equal to the vector sum of the
angular velocities <^2 and <^i at the gears 5 and 7, respectively. Equation
5.7 is analogous to equation 5.2 for the electrical system.
56 CORRECTIVE NETWORKS
Since the same torque fa exists at gears 5 and 7 as the driving point
and, further, since the angular velocity at gear 1 is the vector sum of the
angular velocities at gears 5 and 7, the rotational impedances zr2 and
Ziei at gears 7 and 5, respectively, must be in parallel. That is, the mechan-
ical rotational impedance zet at gear / is
zrt = , 5.8Zri + Zr2
If zr2 is made infinite there can be no motion at gear 5 and therefore
the system behaves the same as if zri were connected to the shaft of
gear 1. In the same way, if Zri is made infinite there is no motion at
gear 7 and the system behaves the same as if Zr2 were connected directly
to the shaft of gear 1.
The acoustical system of Fig. 5.1 consists of a three way connector.
The dimensions are assumed to be small compared to the wavelength.
Therefore, the pressure which actuates the two acoustical impedances is
the same as the driving pressure. The total volume current Xt is the
vector sum of the volume current Xi and X2, that is
Xt = Xi + X2 5.9
Equation 5.9 is analogous to equation 5.2 for the electrical system.
The input acoustical impedance is
zat = , 5.10Zai + Za2
If 2^2 is made infinite, there is no volume current in this branch and of
course Zat becomes Zai. In the same way if z^i is made infinite zat
becomes za2- Thus it will be seen that the acoustical system of Fig. 5.1
is analogous to the electrical system of Fig. 5.1.
5.3. Shunt Corrective Networks
In a shunt corrective electrical network an electrical resistance, induct-
ance, electrical capacitance or a combination of these elements is shunted
across a line.'
^ The term "line" is used to designate an electrical network which prior to the
introduction of the corrective electrical network consisted of a generator in
series with two electrical impedances zei and z^s, termed the input and outputelectrical impedances. The shunt corrective electrical network Ze2 is connectedin parallel with the output electrical impedance 2^3.
SHUNT CORRECTIVE NETWORKS 57
The output current i^ of a line shunted by an electrical network is
given by
eZE2 c 1 1l3 = 5.11
ZjsiZB2 + ZjsiZjis + ZE2ZE3
where zm = input electrical impedance,
ze2 = electrical impedance of the corrective electrical network,
2e3 = output electrical impedance, or the electrical impedance
connected in shunt with the network, and
e = electromotive force in series with the input electrical imped-
ance.
The output velocity x^ of a mechanical rectilineal network which is
analogous to the shunt electrical network is given by
X3 = 5.12ZmiZM2 + ZmiZms + Zm2ZM3
where Zmi = input mechanical rectilineal impedance,
Zm2 = mechanical rectilineal impedance of the corrective mechani-
cal rectilineal network,
zmz = output mechanical rectilineal impedance, and
/m = mechanical driving force in series with the input mechanical
rectilineal impedance.
The output angular velocity 4>s of a mechanical rotational network
which is analogous to the shunt electrical network is given by
4>3=
_^-^'^'^'
_^5.13
ZbiZR2 + ZriZB3 + Zr2ZB3
where Zm — input mechanical rotational impedance,
2jj2 = mechanical rotational impedance of the corrective mechan-
ical rotational network,
2^3 = output mechanical rotational impedance, and
fa = driving torque in series with the input rotational imped-
ance.
58 CORRECTIVE NETWORKS
The output volume current X^ of an acoustical network which is
analogous to the shunt electrical network is given by
X^ = P^5.14
ZaiZA2 + Z.41Z.13 + ZA2ZA3
where zai = input acoustical impedance,
Za2 = acoustical impedance of the corrective network,
2.43 = output acoustical impedance, and
p — driving pressure in series with the input acoustical imped-
ance.
5.4. Inductance in Shunt with a Line and the Mechanical Rectilineal,
Mechanical Rotational and Acoustical Analogies
In Fig. 5.2 an inductance is shunted across a line. The electrical
impedance of an inductance is
ze2 — j<^L 5.15
where co = ItJ,
f = frequency, in cycles per second, and
L = inductance, in abhenries.
Equations 5.11 and 5.15 show that if the electrical impedance of the
inductance is small compared to the input and output electrical imped-
ances, the transmission will be small. If the electrical impedance of the
inductance is large compared to the input and output electrical imped-
ances, the attenuation introduced by the inductance will be negligible.
Since the electrical impedance of an inductance is proportional to the
frequency, the transmission will increase with frequency as shown by the
characteristic * of Fig. 5.2.
The mechanical rectilineal impedance of the mass in Fig. 5.2 is
ZM2 = J<^f^ 5.16
where m = mass, in grams.
When the mass reactance in the mechanical rectilineal system of Fig.
5.2 is small compared to the load or driving mechanical rectilineal imped-
'' The verbal description and the depicted transmission frequency character-
istics in this chapter tacitly assume preponderately resistive input and output
impedances. Of course the equations are valid for any kind of input and out-
put impedances.
INDUCTANCE IN SHUNT WITH A LINE 59
ance, equations 5.12 and 5.16 show that the velocity of the mass will be
relatively large and there will be very little velocity transmitted to the
load. If the mass reactance is comparatively large the mass will remain
practically stationary and the behavior will be the same as a directly
coupled system. Since the mechanical rectilineal impedance of a mass is
proportional to the frequency, the transmission will increase with fre-
quency as shown by the characteristic of Fig. 5.2.
ELECTRICALI
mI/7^
RECTILINEAL
=J L.
^^^^,^^^^u^^^^^^^^^^^^^^^^^^^
ACOUSTICAL
FREQUENCY
ROTATIONAL
MECHANICAL
Fig. 5.2. Inductance in shunt with a hne and the mechanical rectilineal, mechanical
rotational and acoustical analogies. The graph depicts the transmission frequency
characteristic.
The mechanical rotational impedance of the flywheel in Fig. 5.2 is
zk2 =7'^^ 5.17
where / = moment of inertia, in gram (centimeter)^.
When the moment of inertia reactance in the mechanical rotational
system of Fig. 5.2 is small compared to the load or driving mechanical
rotational impedance, equations 5.13 and 5.17 show that the angular
velocity of the flywheel will be relatively large and there will be very
little angular velocity transmitted to the load. If the moment of inertia
reactance is comparatively large the flywheel will remain practically sta-
tionary and the behavior will be the same as a directly coupled system.
60 CORRECTIVE NETWORKS
Since the mechanical rotational impedance of a moment of inertia is
proportional to the frequency, the transmission will increase with fre-
quency as shown by the characteristic of Fig. 5.2.
The acoustical system of Fig. 5.2 consists of a pipe with a side branch
forming an inertance. The acoustical impedance of the inertance in
Fig. 5.2 is
2.42 = joiM 5.18
where M = inertance, in grams per (centimeter)*.
Equations 5.14 and 5.18 show that at low frequencies the acoustical
reactance of the inertance is small compared to the acoustical impedance
of the pipe and the sound is shunted out through the hole. At high fre-
quencies the acoustical reactance of the inertance is large compared to
the acoustical impedance of the pipe and the sound wave flows down the
pipe the same as it would in the absence of a branch. Since the acoustical
impedance of an inertance is proportional to the frequency, the transmis-
sion will increase with frequency as shown by the characteristic of
Fig. 5.2.
5.5. Electrical Capacitance in Shunt with a Line and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies
In Fig. 5.3 an electrical capacitance is shunted across a line. Theelectrical impedance of an electrical capacitance is
2iJ2 = T-— 5.19jwLe
where Ce = electrical capacitance, in abfarads.
The electrical reactance of an electrical capacitance is inversely propor-
tional to the frequency. Therefore, from equations 5.11 and 5.19 the
transmission will decrease with increase in frequency as shown by the
characteristic of Fig. 5.3.
The mechanical rectilineal impedance of the compliance in Fig. 5.3 is
ZM2 = T^ 5.20
where Cm = compliance, in centimeters per dyne.
ELECTRICAL CAPACITANCE IN SHUNT WITH A LINE 61
The mechanical rectilineal reactance of the compliance of the recti-
lineal system, Fig. 5.3, is inversely proportional to the frequency. Equa-
tions 5.12 and 5.20 show that at low frequencies the velocity at the input
to the compliance will be small and the behavior will be practically the
same as that of a directly coupled system. At high frequencies the
velocity of the compliance will be practically the same as the input
iELECTRICAL
rrL^
ACOUSTICAL
ROTATIONAL
MECHANICAL
Fig. 5.3. Electrical capacitance in shunt with a line and the mechanical rectihneal,
mechanical rotational and acoustical analogies. The graph depicts the transmission
frequency characteristic.
velocity and there will be very little velocity transmitted to the load.
The transmission characteristic of this system obtained from equations
5.12 and 5.20 is shown in Fig. 5.3.
The mechanical rotational impedance of the rotational compliance
of Fig. 5.3 is
1
2fi2
i^Cff5.21
where Cr = rotational compliance, in radians per dyne per centimeter.
The mechanical rotational impedance of the rotational compliance of
the mechanical rotational system of Fig. 5.3 is inversely proportional to
the frequency. Equations 5.13 and 5.21 show that at low frequencies
62 CORRECTIVE NETWORKS
the angular velocity at the input to the rotational compliance will be
small and the behavior will be practically the same as that of a directly
coupled system. At high frequencies the angular velocity of the rota-
tional compliance will be the same as the input angular velocity and
there will be very little angular velocity transmitted to the load. The
transmission characteristic of this system obtained from equations 5.13
and 5.21 is shown in Fig. 5.3.
The acoustical system of Fig. 5.3 consists of a pipe with an enlarged
portion forming an acoustical capacitance. The acoustical impedance of
an acoustical capacitance is
Z.12 = ^-^ 5.22
where Ca = acoustical capacitance, in (centimeter)^ per dyne.
At low frequencies the acoustical reactance of the acoustical capaci-
tance is large compared to the impedance of the pipe and the sound
flows down the pipe the same as it would in the absence of the enlarge-
ment. At high frequencies the acoustical reactance of the acoustical
capacitance is small compared to the acoustical impedance of the pipe
and the sound is shunted out by the enlargement. Since the acoustical
reactance is inversely proportional to the frequency, equations 5.14 and
5.22 show that the transmission will decrease with frequency as shown by
the characteristic of Fig. 5.3.
5.6. Inductance and Electrical Capacitance in Series, in Shunt with a
Line and the Mechanical Rectilineal, Mechanical Rotational andAcoustical Analogies
Fig. 5.4 shows an inductance and electrical capacitance connected
in series across a line. The mechanical rectilineal, mechanical rota-
tional, and acoustical analogies are also shown in Fig. 5.4.
The electrical impedance of the electrical network is
ze2 = j^L + -r-^ 5.23jwLe
where L = inductance, in abhenries, and
Ce = electrical capacitance, in abfarads.
The output current can be obtained from equations 5.11 and 5.23.
INDUCTANCE AND ELECTRICAL CAPACITANCE 63
The mechanical rectilineal impedance of the mechanical rectilineal
system is
Zm2 = j<^m + ^r-^ 5.24
where m = mass, in grams, and
Cm = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.12 and 5.24.
ELECTRICAL
^^^^^l^^^^^'-^^'.^
m^^<^'\""^'n^u""^"i"""^^'^^^'
ACOUSTICAL
FREQUENCY
ROTATIONAL
MECHANICAL
Fig. 5.4. Inductance and electrical capacitance in series, in shunt with a line and the
mechanical rectilineal, mechanical rotational and acoustical analogies. The graphdepicts the transmission frequency characteristic.
The mechanical rotational impedance of the mechanical rotational
system is
2fi2 = j^l + ^-^ 5.25
where / = moment of inertia, in gram (centimeter)^, and
Cu = rotational compliance, in radians per dyne per centimeter.
The output angular velocity can be obtained from equations 5.13 and
5.25.
64 CORRECTIVE NETWORKS
The acoustical impedance of the acoustical system is
1
za2 = J^M + T—
r
5.26
where M = inertance, in grams per (centimeter)'', and
Ca = acoustical capacitance, in (centimeter)" per dyne.
The output volume current can be obtained from equations 5.14 and5.26.
At low frequencies the four systems behave the same as those of
Fig. 5.3 and there is very little attenuation. At high frequencies the
systems behave the same as those of Fig. 5.2 and there is very little
attenuation. At the resonant frequency of the inductance and the elec-
trical capacitance the electrical impedance is zero and equations 5.11 and
5.23 show that there is no transmission at this frequency. At the reso-
nant frequency of the mass and compliance no motion is transmitted be-
cause the force required to drive the resonant system is zero. Equations
5.12 and 5.24 also show that there is no transmission at the resonant
frequency of the mass and compliance. At the resonant frequency of the
moment of inertia and rotational compliance no angular motion is trans-
mitted because the torque required to actuate the resonant system is
zero. Equations 5.13 and 5.25 also show that there is no transmission at
the resonant frequency of the moment of inertia and rotational compli-
ance. At the resonant frequency of the inertance and acoustical capaci-
tance there will be no transmission because the pressure at the input to
the resonator is zero. Equations 5.14 and 5.26 also show that there is no
transmission at the resonant frequency of the inertance and acoustical
capacitance. The transmission characteristics of the four systems are
shown in Fig. 5.4.
5.7. Inductance and Electrical Capacitance in Parallel, in Shunt with
a Line and the Mechanical Rectilineal, Mechanical Rotational andAcoustical Analogies
Fig. 5.5 shows an inductance and electrical capacitance connected in
parallel across a line. The mechanical rectilineal, mechanical rotational
and acoustical equivalents are also shown in Fig. 5.5.
INDUCTANCE AND FXECTRICAL CAPACITANCE 65
The electrical impedance of the electrical network of Fig. 5.5 is
jo}L-£2 5.27
where L = inductance, in abhenries, and
Ce = electrical capacitance, in abfarads.
The output current can be obtained from equations 5.1 1 and 5.27.
ELECTRICAL
M
Ikkkkkkkkm^^u^^^'." ^^^'^^'"^"^'^'^""
ACOUSTICAL
FREQUENCY
ROTATIONAL
MECHANICAL
Fig. 5.5. Inductance and electrical capacitance in parallel, in shunt with a line and the
mechanical rectilineal, mechanical rotational and acoustical analogies. The graph
depicts the transmission frequency characteristic.
The mechanical rectilineal impedance of the mechanical rectilineal
system of Fig. 5.5 is
juimZl/2 =
1 - J^mC5.28
M
where m = mass, in grams, and
Cm = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.12 and 5.28.
66 CORRECTIVE NETWORKS
The mechanical rotational impedance of the mechanical rotational
system of Fig. 5.5 is
1 - co^/GeZR2 =
.5.29
where / = moment of inertia, in gram (centimeter)^, and
Cr = rotational compliance, in radians per dyne per centimeter.
The output angular velocity can be obtained from equations 5.13 and
5.29.
The acoustical impedance of the acoustical system of Fig. 5.5 is
joiM''' =
1 - .^MC,^-'^
where M = inertance, in grams per (centimeter)*, and
Ca = acoustical capacitance, in (centimeter)^ per dyne.
The output volume current can be obtained from equations 5.14 and
5.30.
At low frequencies the systems behave the same as those of Fig. 5.2
and the transmission is small. At high frequencies the systems behave
the same as those of Fig. 5.3 and the transmission is again small. At the
resonant frequency of the inductance and electrical capacitance the elec-
trical impedance is infinite and equations 5.11 and 5.27 show that the
shunt circuit introduces no attenuation at the resonant frequency. Atthe resonant frequency of the mass and compliance the input to the
spring does not move because the mechanical rectilineal impedance is
infinite and the behavior is the same as a directly coupled system. Equa-tions 5.12 and 5.28 also show that there is no attenuation due to the
shunt system at the resonant frequency of the mass and compliance. Atthe resonant frequency of the moment of inertia with the rotational com-
pliance the input to the spring will not turn because the mechanical rota-
tional impedance is infinite and the behavior is the same as a directly
coupled system. Equations 5.13 and 5.29 also show that there is no
attenuation due to the shunt system at the resonant frequency. At the
resonant frequency of the inertance and acoustical capacitance the input
volume current is zero because the acoustical impedance is infinite and
the behavior is the same as a plain pipe. Equations 5.14 and 5.30 also
show that there is no attenuation due to the shunt system at the reso-
THREE SERIES ELEMENTS 67
nant frequency. The transmission characteristics of the four systems
are shown in Fig. 5.5.
5.8. Electrical Resistance, Inductance and Electrical Capacitance in
Series, in Shunt with a Line and the Mechanical Rectilineal,
Mechanical Rotational and Acoustical Analogies
Fig. 5.6 shows an electrical resistance, inductance and electrical capac-
itance in series, in shunt with a line. The mechanical rectilineal, mechan-
ELECTRICAL
tkmu.^^rr
^^^^^ss^ A.min.tim
ACOUSTICAL
ROTATIONAL
MECHANICAL
Fig. 5.6. Electrical resistance, inductance and electrical capacitance in series, in shunt
with a line and the mechanical rectilineal, mechanical rotational and acoustical analogies.
The graph depicts the transmission frequency characteristic.
ical rotational and acoustical analogies are also shown in Fig. 5.6. Theelectrical impedance of the electrical network is
ze2 = rE -\- joiL + ^-— 5.31J(X:Le
where rg = electrical resistance, in abohms,
L = inductance, in abhenries, and
Ce = electrical capacitance, in abfarads.
The output current can be obtained from equations 5.11 and 5.31.
68 CORRECTIVE NETWORKS
The mechanical rectilineal impedance of the mechanical rectilineal
system is
zjf2 = ru + ji^m + . 5.32
where Vm = mechanical rectilineal resistance, in mechanical ohms,m = mass, in grams.
Cm = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.12 and 5.32.
The mechanical rotational impedance of the mechanical rotational
system is
r 1
2«2 = r^j + jo^I + ^^r 5.33
where vr = mechanical rotational resistance, in rotational ohms,
/ = moment of inertia, in gram (centimeter)^, and
Cr — rotational compliance, in radians per dyne per centimeter.
The output rotational velocity can be obtained from equations 5.13
and 5.33.
The acoustical impedance of the acoustical system is
1
ZA2 = rA-\- J^M + -T-— 5.34JicLa
where va = acoustical resistance, in acoustical ohms,
M = inertance, in grams per (centimeter)*, and
Ca = acoustical capacitance, in (centimeter)^ per dyne.
The output volume current can be obtained from equations 5.14 and5.34.
At low frequencies the systems behave the same as those of Fig. 5.3
and there is very little attenuation. At high frequencies the systems
behave the same as those of Fig. 5.2 and there is very little attenuation.
At the resonant frequency of the inductance and electrical capacitance
the electrical reactance is zero. Therefore, from equations 5.11 and 5.31
the transmission as Influenced by the network is governed by the elec-
trical resistance. At the resonant frequency of the mass and compliance
the mechanical rectilineal reactance is zero. Therefore, from equations
5.12 and 5.32 the transmission as influenced by the mechanical rectilineal
THREE PARALLEL ELEMENTS 69
system is governed by the mechanical rectihneal resistance. At the
resonant frequency of the moment of inertia and rotational compliance
the mechanical rotational reactance is zero. Therefore, from equations
5.13 and 5.33 the transmission as influenced by the mechanical rotational
system is governed by the mechanical rotational resistance. At the
resonant frequency of the inertance and acoustical capacitance the
acoustical reactance is zero. Therefore, from equations 5.14 and 5.34
the transmission as influenced by the acoustical system is governed by
the acoustical resistance. The transmission characteristic of these sys-
tems is shown in Fig. 5.6. This characteristic at the high and low fre-
quencies is the same as that of Fig. 5.4. However, in the region of reso-
nance the resistance in each of the four systems decreases the attenua-
tion as depicted by the transmission characteristic of Fig. 5.6.
5.9. Electrical Resistance, Inductance and Electrical Capacitance in
Parallel, in Shunt with a Line and the Mechanical Rectilineal,
Mechanical Rotational and Acoustical Analogies
Fig. 5.7 shows an electrical resistance, inductance and electrical capaci-
tance connected m parallel across a hne. The mechanical rectihneal,
mechanical rotational and acoustical analogies are also shown in Fig. 5.7.
The electrical impedance of the electrical network is
2^2 = 2—rn ^ T
^--^^
Ve — ^ rELuLE + Jf^L,
where Ve = electrical resistance, in abohms,
L = inductance, in abhenries, and
Ce = electrical capacitance, in abfarads.
The output current can be obtained from equations 5.11 and 5.35.
The mechanical rectilineal impedance of the mechanical system is
2.1/2 = 2 n IT^ ^"^°
where rjf = mechanical rectilineal resistance, in mechanical ohms,
m = mass, in grams, and
Cm = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.12 and 5.36.
70 CORRECTIVE NETWORKS
The mechanical rotational impedance of the rotational system is
jo^rRlZR2 5.37
rn — oi^rRlCn + jwl
where ru = mechanical rotational resistance, in rotational ohms,
/ = moment of inertia, in gram (centimeter)^, and
Cb, = rotational compliance, in radians per dyne per centimeter.
The output rotational velocity can be obtained from equations 5.13
and 5.37.
ELECTRICAL
Bsssssssm Mlk^UU<U^kkU
ACOUSTICAL
ROTATIONAL
MECHANICAL
Fig. S.7. Electrical resistance, inductance and electrical capacitance in parallel, in shunt
with a line and the mechanical rectilineal, mechanical rotational and acoustical analogies.
The graph depicts the transmission frequency characteristic.
The acoustical impedance of the acoustical system is
JuvaM5.38
rA — coViMCa + jwMwhere r^ = acoustical resistance, in acoustical ohms,
M = inertance, in grams per (centimeter)'', and
Ca = acoustical capacitance, in (centimeter)'^ per dyne.
The output volume current can be obtained from equations 5.14 and5.38.
SERIES CORRECTIVE NETWORKS 71
At low frequencies the systems behave the same as those of Fig. 5.2
and the transmission is small. At high frequencies the systems behave
the same as those of Fig. 5.3 and the transmission is again small. At the
resonant frequency the electrical reactance is infinite and therefore from
equations 5.11 and 5.35 the attenuation is due to the electrical resistance.
At the resonant frequency of the mass and compliance, Fig. 5.7, the
mechanical rectilineal reactance is infinite and therefore from equations
5.12 and 5.36 the attenuation is due to the mechanical rectilineal resist-
ance. At the resonant frequency of the moment of inertia and rotational
compliance. Fig. 5.7, the mechanical rotational reactance is infinite and
from equations 5.13 and 5.37 the attenuation is due to the mechanical
rotational resistance. At the resonant frequency of the inertance and
acoustical capacitance, Fig. 5.7, the acoustical reactance is infinite and
from equations 5.14 and 5.38 the attenuation is due to the acoustical
resistance. The transmission characteristic of these systems is shown in
Fig. 5.7. This characteristic at the low and high frequencies is the same
as that of Fig. 5.5. However, in the region of resonance the resistance in
each of the four systems introduces attenuation as depicted by the
transmission characteristic of Fig. 5.7.
5.10. Series Corrective Networks
In a series electrical network an electrical resistance, inductance, elec-
trical capacitance or a combination of these elements is connected in
series with a line.^
The output current i^ of a line containing a series electrical network is
given by
/3= -^—r- 5.39
Zei + Ze2 + Zil'S
where zei = input electrical impedance,
ze2 = electrical impedance of the corrective electrical network,
Ze3 = output electrical impedance, and
e = electromotive force in series with the three electrical im-
pedances.
° The term "line" is used to designate an electrical network which prior to
the introduction of the corrective electrical network consisted of a generator in
series with two electrical impedances Zei and zes, termed the input and outputelectrical impedances. The series corrective electrical network Ze2 is connectedin series with the input and output electrical impedances z^i and Zes.
72 CORRECTIVE NETWORKS
The output velocity x^, of a mechanical rectilineal network which is
analogous to the series electrical network is given by
*3 = — 5.40Zilfl + Zil/2 + ZM2,
where Zmi = input mechanical rectilineal impedance,
Zm2 = mechanical rectilineal impedance of the corrective mechan-
ical rectilineal network,
ZAf3 = output mechanical rectilineal impedance, and
Jm = mechanical driving force in series with mechanical imped-
ances.
The output angular velocity 03 of a mechanical rotational network
which is analogous to the series electrical network is given by
'J>3= — 5.41
ZRI + Z/S2 + Z/J3
where Zri = input mechanical rotational impedance,
2K2 = mechanical rotational impedance of the corrective mechani-
cal rotational network,
ZB3 = output mechanical rotational impedance, and
Jr = rotational driving torque in series with the mechanical
impedances.
The output volume current X^ of an acoustical network which is
analogous to the series electrical network is given by
PX3 = 5.42
Z-U + Z_12 + Zas,
where 2.41 = input acoustical impedance,
za2 — acoustical impedance of the corrective acoustical network,
2.43 = output acoustical impedance, and
p = driving pressure in series with the acoustical impedances.
5.11. Inductance in Series with a Line and the Mechanical Rectilineal,
Mechanical Rotational and Acoustical Analogies
In Fig. 5.8 an inductance is connected in series with a line. The elec-
trical impedance of the inductance is
ze2 = j^L, 5.43
where L = inductance, in abhenries.
INDUCTANCE IN SERIES WITH A LINE 73
Equations 5.39 and 5.43 show that if the electrical impedance of the
inductance is small compared to the input and output electrical imped-
ances the attenuation introduced by the inductance will be small. If the
electrical impedance of the inductance is large compared to the input
and output electrical impedances the current transmission will be small.
Since the electrical impedance of an inductance is proportional to the
frequency, the transmission will decrease with frequency as shown in
Fie. 5.8.
-nsim—RECTILINEAL
# FREQUENCY
ACOUSTICAL ROTATIONAL
MECHANICAL
Fig. 5.8. Inductance in series with a line and the mechanical rectilineal, mechanical
rotational and acoustical analogies. The graph depicts the transmission frequency
characteristic.
The mechanical rectilineal impedance of the mass in Fig. 5.8 is
zm2 = j^m 5.44
where tn = mass, in grams.
In the mechanical rectilineal system of Fig. 5.8 equations 5.40 and
5.44 show that if the mass reactance is small compared to the load or
driving mechanical rectilineal impedance the addition of the mass will
cause very little reduction in the velocity transmitted to the load. If
the mass reactance is comparatively large the mass will remain practi-
cally stationary and the velocity transmitted to the load will be small.
Since the mechanical rectilineal impedance of a mass is proportional to
the frequency, the transmission will decrease with frequency as shown by
the characteristic of Fig. 5.8.
The mechanical rotational impedance of the flywheel in Fig. 5.8 is
zji2 = jo)I 5.45
where / = moment of inertia, in gram (centimeters)^.
74 CORRECTIVE NETWORKS
In the mechanical rotational system of Fig. 5.8 equations 5.41 and 5.45
show that if the moment of inertia reactance is small compared to the
load or driving mechanical rectilineal impedance the addition of the mo-ment of inertia will cause very little reduction in the angular velocity
transmitted to the load. If the moment of inertia reactance is compara-
tively large the flywheel will remain practically stationary and the veloc-
ity transmitted to the load will be small. Since the mechanical rotational
impedance of a moment of inertia is proportional to the frequency, the
transmission will decrease with frequency as shown by the characteristic
of Fig. 5.48.
The acoustical system of Fig. 5.8 consists of a pipe with a constriction
which forms an inertance. The acoustical impedance of the inertance in
Fig. 5.8 is
za2 = jo^M 5.46
where M = inertance, in grams per (centimeter)*.
Equations 5.42 and 5.46 show that at the low frequencies where the
acoustical impedance of an inertance is small compared to input and out-
put acoustical impedances the attenuation introduced by the inertance
is small. At the high frequencies the acoustical impedance of the inert-
ance is large and the transmission is small. Since the acoustical imped-
ance of an inertance is proportional to the frequency the transmission will
decrease with frequency as shown by the characteristic of Fig. 5.8.
5.12. Electrical Capacitance in Series with a Line and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies
In Fig. 5.9 an electrical capacitance is connected in series with a line.
The electrical impedance of the electrical capacitance of Fig. 5.9 is
2£2 = r- -5.47jwLe
where Ce = electrical capacitance, in abfarads.
Equation 5.39 shows that if the electrical impedance of the electrical
capacitance is large compared to the input or output electrical imped-
ances, the attenuation introduced by the electrical capacitance will be
large. If the electrical impedance of the electrical capacitance is small
compared to the input and output electrical impedances the attenuation
ELECTRICAL CAPACITANCE IN SERIES WITH A LINE 75
will be small. Since the electrical impedance of an electrical capacitance
is inversely proportional to the frequency, as shown by equation 5.47,
the transmission will increase with frequency as shown in Fig. 5.9.
The mechanical rectilineal impedance of Fig. 5.9 is
1
Z.l/2 =
where Cm — compliance, in centimeters per dyne.
5.48
ELECTRICAL777^77 777777
RECTILINEAL
FREQUENCY
ACOUSTICAL ROTATIONAL
MECHANICAL
Fig. 5.9. Electrical capacitance in series with a line and the mechanical rectihneal,
mechanical rotational and acoustical analogies. The graph depicts the transmission
frequency characteristic.
Equation 5.48 shows that the mechanical impedance of the compliance
of the mechanical rectilineal system of Fig. 5.9 is inversely proportional
to the frequency. Equations 5.40 and 5.48 show that at low frequencies
the input velocity to the compliance is relatively small and there will be
little transmission. At high frequencies the input velocity to the compli-
ance is relatively large and therefore it introduces very little impedance
to motion. Therefore the transmission characteristic will be shown in
Fig. 5.9.
The mechanical rotational impedance of Fig. 5.9 is
^R21
J^Cr5.49
where Cr = rotational compliance, in radians per dyne per centimeter.
76 CORRECTIVE NETWORKS
Equation 5.49 shows that the mechanical rotational impedance of the
rotational compliance of the mechanical rotational system of Fig. 5.9 is
inversely proportional to the frequency. Equations 5.41 and 5.48 showthat at the low frequencies the input angular velocity to the rotational
compliance is relatively small and the transmission is low. At the high
frequencies the angular velocity of the rotational compliance is relatively
large and it introduces very little impedance to motion. Therefore the
transmission characteristic will be as shown in Fig. 5.9.
There is no simple purely acoustical system which is analogous to an
electrical capacitance in series with a line. The analogy shown in Fig.
5.9 consists of a stiffness controlled diaphragm, that is, the mass of the
diaphragm is small and the stiffness high so that the frecjuency range
under consideration is well below the natural resonant frequency of the
diaphragm and suspension. The acoustical capacitance of this system is
Ca = CyS' 5.50
where Ca = acoustical capacitance, in (centimeter)" per dyne.
Cm = compliance of the suspension system, in centimeters per
dyne, and
S = area of the diaphragm, in square centimeters.
The acoustical impedance of Fig. 5.9 is
2.42 = -r-— 5.51JwCa
where Ca = acoustical capacitance of equation 5.50.
It will be seen that this system will not transmit a steady flow of a gas
in the same way that the electrical circuit of Fig. 5.9 will not transmit
direct current. Equation 5.51 shows that the acoustical impedance of an
acoustical capacitance is inversely proportional to the frequency. Equa-
tions 5.42 and 5.51 show that the transmission will increase with increase
of the frequency as shown by the characteristic of Fig. 5.9.
5.13. Inductance and Electrical Capacitance in Series with a Line and
the Mechanical Rectilineal, Mechanical Rotational, and Acous-
tical Analogies
Fig. 5.10 shows an inductance and acoustical capacitance connected in
series with a line. The mechanical rectilineal, mechanical rotational and
acoustical analogies are also shown in Fig. 5.10.
INDUCTANCE AND ELECTRICAL CAPACITANCE 77
The electrical impedance of the electrical network of Fig. 5.10 is
1
ZJ52 = j^LjwCe
5.52
where L = inductance, in abhenries, and
Ce = electrical capacitance, in abfarads.
The output current can be obtained from equations 5.39 and 5.52.
WS^
ELECTRICAL/77777 ^77777
RECTILINEAL
FREQUENCY
ACOUSTICAL ROTATIONAL
MECHANICAL
Fig. 5.10. Inductance and electrical capacitance in scries with a line and the mechanical
rectilineal, mechanical rotational and acoustical analogies. The graph depicts the
transmission frequency characteristic.
The mechanical rectilineal impedance of the mechanical rectilineal
system of Fig. 5.10 is
2.1/2 = j<^m + T-—- 5.53
where m = mass, in grams, and
C_if = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.40 and 5.53.
The mechanical rotational impedance of the mechanical rotational
system of Fig. 5.10 is
ze2 = i^I + ^^~ 5.54
where / = moment of inertia, in gram (centimeter)^, and
Cr = rotational compliance, in radians per dyne per centimeter.
78 CORRECTIVE NETWORKS
The output angular velocity can be obtained from equations 5.41 and
5.54.
The acoustical impedance of the acoustical system of Fig. 5.10 is
ZA2 = j^M + ^-7- 5.55jwLa
where M = inertance, in grams per (centimeter)*, and
Ca = acoustical capacitance, in (centimeter)^ per dyne.
The expression for the acoustical capacitance is given by equation 5.50.
The output volume current can be obtained from equations 5.42 and
5.55.
At low frequencies the four systems behave the same as those of Fig.
5.9 and the transmission is small. At high frequencies the systems
behave the same as those of Fig. 5.8, and again the transmission is small.
At the resonant frequency of the inductance and electrical capacitance
the electrical impedance of these two elements in series is zero and equa-
tions 5.39 and 5.52 show that the series circuit introduces no attenuation.
At the resonant frequency of the mass and compliance the force required
to drive the resonant system is zero and therefore the attenuation intro-
duced by the system is zero. Equations 5.40 and 5.53 also show that the
attenuation is zero at the resonant frequency of the mass and compliance.
At the resonant frequency of the moment of inertia and rotational com-
pliance the torque required to drive the resonant system is zero and
therefore the attenuation introduced by the system is zero. Equations
5.41 and 5.54 also show that the attenuation is zero at the resonant fre-
quency of the moment of inertia and rotational compliance. At the reso-
nant frequency of the inertance and acoustical capacitance the pressure
required to actuate the resonant system is zero. Equations 5.42 and 5.55
also show that the attenuation introduced by the system is zero. Thetransmission characteristics of the four systems are shown in Fig. 5.10.
5.14. Inductance and Electrical Capacitance in Parallel, in Series with
a Line and the Mechanical Rectilineal, Mechanical Rotational
and Acoustical Analogies
Fig. 5.11 shows an inductance and electrical capacitance in parallel
connected in series with a line. The mechanical rectilineal, mechanical
rotational and acoustical analogies are also shown in Fig. 5.11.
INDUCTANCE AND ELECTRICAL CAPACITANCE 79
The electrical impedance of the electrical network of Fig. 5.11 is
ZE21 - co^LCe
5.56
where L = inductance, in abhenries, and
Ce = electrical capacitance, in abfarads.
The output current can be obtained from equations 5.39 and 5.56.
^7^-
ELECTRICAL
ymUULUWTO'
ACOUSTICAL ROTATIONAL
MECHANICAL
Fig. 5.11. Inductance and electrical capacitance in parallel, in series with a line and the
mechanical rectihneal, mechanical rotational and acoustical analogies. The graph
depicts the transmission frequency characteristic.
The mechanical rectilineal impedance of the mechanical rectilineal
system of Fig. 5.11 is
2.1/2 =1 — (j?mCM
S.Sl
where m = mass, in grams, and
Cm = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.40 and 5.57.
The mechanical rotational impedance of the mechanical rotational
system of Fig. 5.11 is
2/22 = :, 2 7-" •'••''^
1 w-/Cfl
80 CORRECTIVE NETWORKS
where / = moment of inertia, in grams (centimeter)^, and
Cr = rotational compliance, in radians per dyne per centimeter.
The output angular velocity can be obtained from equations 5.41 and
5.58.
The acoustical impedance of the acoustical system of Fig. 5.11 is
1 - a)2MCi2.42 == -. ^^jv^ 5.59
where M = inertance, in grams per (centimeter)*, and
Ca = acoustical capacitance, in (centimeter)'^ per dyne.
The output volume current can be obtained from equations 5.42 and
5.59.
At low frequencies the systems behave the same as those of Fig. 5.8
and the attenuation is small. At high frequencies the systems behave
the same as those of Fig. 5.9 and again the attenuation is small. At the
resonant frequency of the inductance and electrical capacitance the elec-
trical impedance is infinite and equations 5.39 and 5.56 show that there
is no transmission. At the resonant frequency of the mass and compli-
ance of Fig. 5.11 the mechanical rectilineal impedance is infinite and the
input to the spring does not move. Equations 5.40 and 5.57 show that
there is no transmission at this frequency. At the resonant frequency of
the moment of inertia and rotational compliance of Fig. 5.11 the mechan-
ical rotational impedance is infinite and the input to the spring does not
rotate. Equations 5.41 and 5.58 show that there is no transmission at
this frequency. At the resonant frequency of the inertance and acousti-
cal capacitance of Fig. 5.11 equation 5.59 shows that the acoustical imped-
ance is infinite. Equation 5.42 shows that there is no transmission at
this frequency. The transmission characteristics of the four systems are
shown in Fig. 5.11.
5.15. Electrical Resistance, Inductance and Electrical Capacitance in
Series with a Line and the Mechanical Rectilineal, Mechanical
Rotational and Acoustical Analogies
Fig. 5.12 shows an electrical resistance, inductance and electrical
capacitance in series with a line. The mechanical rectilineal, mechanical
rotational and acoustical analogies are also shown in Fig. 5.12.
THREE SERIES ELEMENTS 81
The electrical impedance of the electrical network of Fig. 5.12 is
Z£2 = rE + jwL + TJ^Ce
where rj? = electrical resistance, in abohms,
L = inductance, in abhenries, and
Ce = capacitance, in abfarads.
The output current can be obtained from equations 5.39 and 5.60.
c«
5.60
^vw^MHI
ELECTRICAL
FREQUENCY
ACOUSTICAL POTATIONAL
MECHANICAL
Fic. 5.12. Electrical resistance, inductance and electrical capacitance in series with a line
and the mechanical rectilineal, mechanical rotational and acoustical analogies. 'J'hc
graph depicts the transmission frequency characteristic.
The mechanical rectilineal impedance of the mechanical system of
Fig. 5.12 is
Z)/2 = rM + jwm + . 5.61
where Tm = mechanical rectilineal resistance, in mechanical ohms,
m = mass, in grams,
Cm = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.40 and 5.61.
The mechanical rotational impedance of the mechanical rotational
system is
Zfl2 = Tij + jwl + T-^r 5.62
82 CORRECTIVE NETWORKS
where vr = mechanical rotational resistance, in rotational ohms,
/ = moment of inertia, in gram (centimeter)^, and
Cr = rotational compliance, in radians per dyne per centimeter.
The output rotational velocity can be obtained from equations 5.41
and 5.62.
The acoustical impedance of the acoustical system of Fig. 5.12 is
2.42 = TA -[-jwM + -T-z^ 5.63_;coC.4
where va = acoustical resistance, in acoustical ohms,
M = inertance, in grams per (centimeter)*, and
Ca = acoustical capacitance, in (centimeter)^ per dyne.
The output volume current can be obtained from equations 5.42 and
5.63.
At low frequencies the four systems behave the same as those of Fig.
5.9 and the transmission is small. At high frequencies the systems be-
have the same as those of Fig. 5.8, and again the transmission is small.
At the resonant frequency of the inductance and electrical capacitance
the series electrical reactance is zero. Therefore, from equations 5.39
and 5.60 the attenuation is due to the electrical resistance. At the reso-
nant frequency of the mass and compliance. Fig. 5.12, the mechanical
rectilineal reactance is zero. Therefore, from equations 5.40 and 5.61 the
attenuation is due to the mechanical resistance. At the resonant fre-
quency of the moment of inertia and the rotational compliance, Fig.
5.12, the mechanical rotational reactance is zero. Therefore, from equa-
tions 5.41 and 5.62 the attenuation is due to the mechanical rotational
resistance. At the resonant frequency of the inertance and acoustical
capacitance. Fig. 5.12, the acoustical reactance is zero. Therefore, from
equations 5.42 and 5.63 the attenuation is due to the acoustical resist-
ance. The transmission characteristic of these systems is shown in Fig.
5.12. This characteristic at the low and high frequencies is the same as
that of Fig. 5.10. However, in the region of resonance the resistance in
each of the four systems introduces attenuation as depicted by the trans-
mission characteristic of Fig. 5.12.
THREE PARALLEL ELEMENTS 83
5.16. Electrical Resistance, Inductance and Electrical Capacitance in
Parallel, in Series with a Line and the Mechanical Rectilineal,
Mechanical Rotational and Acoustical Analogies
Fig. 5.13 shows an electrical resistance, inductance and electrical
capacitance in parallel in series with a line. The mechanical rectilineal,
L
mmAAA/-
HHELECTRICAL
ACOUSTICALROTATIONAL
MECHANICAL
Fig. 5.13. Electrical resistance, inductance and electrical capacitance in parallel, in series
with a line and the mechanical rectihneal,mechanical rotational and acoustical analogies.
The graph depicts the transmission frequency characteristic.
mechanical rotational and acoustical analogies are also shown in Fig.
5.13. The electrical impedance of the electrical network of Fig. 5.13 is
ZE2jcorEL
te veLCe + j<^L5.64
where Ve = electrical resistance, in abohms,
L = inductance, in abhenries, and
Ce = electrical capacitance, in abfarads.
The output current can be obtained from equations 5.39 and 5.64.
The mechanical rectilineal impedance of the mechanical rectilineal
system of Fig. 5.13 is
jw7'M inZM2 =
r.M <^ KmC]\j -\- jwm5.65
84 CORRECTIVE NETWORKS
where tm = mechanical rectihneal resistance, in mechanical ohms,
m = mass, in grams, and
Cm = compliance, in centimeters per dyne.
The output velocity can be obtained from equations 5.40 and 5.65.
The mechanical rotational impedance of the mechanical rotational
system of Fig. 5.13 is
J^rglZe2 = 2
—j^—
r^^^T ^-^^rn - 0) rRlLB + jwl
where vr = mechanical rotational resistance, in rotational ohms,
/ = moment of inertia in gram (centimeter)^, and
Cr = rotational compliance, in radians per dyne per centimeter.
The output rotational velocity can be obtained from equations 5.41
and 5.66.
The acoustical impedance of the acoustical system of Fig. 5.13 is
jo^rAMza2 = ~, 5.67
where r^ = acoustical resistance, in acoustical ohms,
M = inertance, in grams per (centimeter)*, and
Ca = acoustical capacitance, in (centimeter)® per dyne.
The output volume current can be obtained from equations 5.42 and
5.67.
At low frequencies the systems behave the same as those of Fig. 5.8
and the attenuation is small. At high frequencies the systems behave the
same as those of Fig. 5.9 and the attenuation is small. At the resonant
frequency of the inductance and electrical capacitance the electrical
reactance is infinite and, therefore, from equations 5.39 and 5.64 the
attenuation is due to the electrical resistance. At the resonant frequency
of the mass and compliance. Fig. 5.13, the mechanical rectilineal react-
ance is infinite. Therefore, from equations 5.40 and 5.65 the attenuation
is due to the mechanical rectilineal resistance. At the resonant frequency
of the moment of inertia and the rotational compliance. Fig. 5.13, the
mechanical rotational impedance is infinite. Therefore, from equations
5.41 and 5.66 the attenuation is due to the mechanical rotational resist-
ance. At the resonant frequency of the inertance and acoustical capaci-
tance. Fig. 5.13, the acoustical reactance is infinite. Therefore, from
SERIES RESISTANCE NETWORK 85
equations 5.42 and 5.67 the attenuation is due to the acoustical resist-
ance. The transmission characteristic of these systems is shown in Fig.
5.13. This characteristic at the low and high frequencies is the same as
that of Fig. 5.11. However, in the region of resonance the resistance in
each of the four systems decreases the attenuation as depicted by the
transmission characteristic of Fig. 5.13.
5.17. Resistance Networks
The use of resistance networks in electrical circuits is well known.
Series and shunt networks are employed to introduce dissipation or
attenuation in the electrical circuits. In the same way mechanical and
acoustical resistance may be used in these systems to introduce dissipa-
tion, damping or attenuation. "T" and "tt" networks are a combination
of series and shunt elements usually employed to introduce attenuation
without mismatching impedances or for matching dissimilar impedances.
5.18. Electrical Resistance in Series with a Line and the Mechanical
Rectilineal, Mechanical Rotational, and Acoustical Analogies
Fig. 5.14 shows an electrical resistance in series with a line. Referring
to equation 5.39 it will be seen that the attenuation will be greater as the
86 CORRECTIVE NETWORKS
greater as the sliding resistance on the brake wheel is made larger. Theacoustical system of Fig. 5.14 shows a system of slits in series with input
and output acoustical impedances. Equation 5.42 shows that the atten-
uation in this system will increase as the acoustic resistance is madelarger.
5.19. Electrical Resistance in Shunt with a Line and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies
Fig. 5.15 shows an electrical resistance in shunt with a line. Referring
to equation 5.11 it will be seen that the attenuation in this case will be
ELECTRICAL
7^V /77777
RECTILINEAL
'^^^^^'^^^^^^"t^'^^^^^'
ACOUSTICAL
ROTATIOMAL
MECHANICAL
Fig. 5.15. Electrical resistance in shunt with a hne and the mechanical rectilineal,
mechanical rotational and acoustical analogies.
greater as the electrical resistance is made smaller. Equation 5.12 showsthat in the same way the attenuation in the mechanical rectilineal system
of Fig. 5.15 will be greater as the sliding resistance is made smaller.
Equation 5.13 shows that the attenuation in the mechanical rotational
system of Fig. 5.15 will be greater as the sliding resistance on the brake
wheel is made smaller. Equation 5.14 shows that in the acoustical sys-
tem of Fig. 5.15 the attenuation will increase as the shunt acoustical
resistance is made smaller.
TYPE RESISTANCE NETWORK 87
5.20. "T" Type Electrical Resistance Network and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies
A "T" type electrical network is shown in Fig. 5.16. Equation 5.11 is
applicable by considering r^i to be added to Zei and rgi to be added to
ze3- fE2 is Ze2 in equation 5.11. In the same way equations 5.12, 5.13
and 5.14 apply to the mechanical rectihneal, mechanical rotational, and
ELECTRICAL
Tmi
//y}}y/7 ^Z}}?}}/'
11
W
IL_
RECTILINEAL
AUkkkkkUkUkkkkUl
ACOUSTICAL
ROTATIONAL
MECHANICAL
Fig. 5.16. "T" type electrical resistance network and the mechanical rectilineal, mechani-
cal rotational and acoustical analogies.
acoustical systems of Fig. 5.16, wherein zm\, zri, and zai is the sum of
^Mi, rm, and r^i and the input impedances, respectively, and Zmz, "Zrz,
and 2^3 is the sum of rui, rui, and r.41 and the output impedances,
respectively.
5.21. *V Type Electrical Resistance Network and the Mechanical
Rectilineal, Mechanical Rotational and Acoustical Analogies
A "it" type electrical network is shown in Fig. 5.17. The "x" type of
electrical network may be used for the same purpose as the "T" network
of the preceding section. The mechanical rectilineal, mechanical rota-
tional, and acoustical resistance systems equivalent to the electrical "x"
88 CORRECTIVE NETWORKS
network are shown in Fig. 5.17. Equation 5.11 may be used to predict
the performance of the electrical system. In this case zei is the input
impedance, ze2 is rE2, and zes is tei in series with rE2 in parallel with the
output impedance. This of course determines only the current in zes-
This current is equal to the vector sum of the current in rE2 and the
ELECTRICAL
^^^llll^JII
ACOUSTICAL
^M2
//}}J///
/77777 77^y^T7
RECTILINEAL
ROTATIONAL
MECHANICAL
Fig. 5.17. "x" type electrical resistance network and the mechanical rectihneal,
mechanical rotational and acoustical analogies.
output impedance. The performance of the mechanical rectilineal, me-
chanical rotational, and acoustical systems may be obtained by similar
considerations employing equations 5.12, 5.13 and 5.14.
5.22. Electrical, Mechanical Rectilineal, Mechanical Rotational and
Acoustical Transformers
A transformer is a transducer used for transferring between two imped-
ances of different values without appreciable reflection loss. Electrical,
mechanical rectilineal, mechanical rotational and acoustical transform-
ers are shown in Fig. 5.18.
TRANSFORMERS 89
In the ideal electrical transformer of Fig. 5.18 the electromotive force,
current and electrical impedance ratios on the two sides of the trans-
former are
N2e-i M e\ 5.68
ZE2 -~ (0 ZBI
5.69
5.70
where A^i = number of turns in the primary, and
A'2 = number of turns on the secondary.
ei, ii, and zei represent the electromotive force, current and electrical
impedance on the primary side and ^2, ^2 and 2^2 represent the electro-
motive force current and electrical impedance on the secondary side.
ACOUSTICALROTATIONAL
MECHANICAL
Fig. 5.18. Electrical, mechanical rectilineal, mechanical rotational and acoustical
transformers.
The mechanical rectilineal transformer of Fig. 5.18 consists of a
rigid massless lever with frictionless bearings. The force, velocity and
90 CORRECTIVE NETWORKS
mechanical rectilineal impedance ratios on the two sides of the lever are
Jmi = J/mi 5.71'2
X2 = — xi 5.72
2^2 = \j) ^Mi 5.73
/i and 4 are the lengths of the lever arms depicted in Fig. 5.18. /mi, •*!
and 23/1 andyjif2) '*2 and 23/2 represents the force, velocity and mechanical
rectilineal impedance on the two sides of the mechanical transformer.
The mechanical rotational transformer, of Fig. 5.18, consists of a mass-
less rigid gear train. The torque, angular velocity and mechanical rota-
tional impedance on the two sides of the gear train are
fR2 = fAi 5.74
4>2 = ik 5.75«2
2/J2 = I T i '^R^ 5.76
di and ^2 are the diameters of the gears depicted in Fig. 5.18. /ri, <t>i
and Ziji and//s2) <t>2 and 27^2 represents the torque, angular velocity and
mechanical rotational impedance on the two sides of the rotational trans-
former.
The acoustical transformer consists of two rigid massless diaphragms
with negligible suspension stiffness coupled together as shown in Fig.
5.18. The pressure, volume current and acoustical impedance on the
two sides of the diaphragm combination are
P2=^pi 5.77
SoX2 = -? Xi 5.78
2^2 = I V J =2^1 5.79
ACOUSTICAL TRANSFORMERS 91
Si and >S'2 are the areas of the two diaphragms, pi, Xi and 2.41 and p2,
X2 and za2 represents the pressure, volume current and acoustical imped-
ance on the two sides of the acoustical transformer.
The acoustical transformer of Fig. 5.18 is not purely an acoustical sys-
tem since it uses mechanical elements in the form of diaphragms. In an
acoustical system a horn may be used to transfer from one impedance to
another impedance of a different value without appreciable reflection
loss. As a matter of fact a horn ^ may be looked upon as an acoustical
transformer, transforming large pressures and small volume currents to
small pressures and large volume currents or the reverse process.
^ For these and other properties of horns see Olson, "Elements of Acoustical
Engineering," D. Van Nostrand Co., New York, 1940.
CHAPTER VI
WAVE FILTERS
6.1. Introduction
The essential function of a wave filter is to let pass desired frequency
bands and to highly attenuate neighboring undesired frequency bands.
An electrical filter is a general type of electrical network in which a
number of recurrent electrical impedance elements are assembled to
form a recurrent structure. Electrical networks of this sort are called
electrical wave filters/ as they pass certain frequencies freely and stop
others. Wave filters analogous to electrical wave filters may be de-
veloped and employed in any wave motion system. j'\coustical ^ and
mechanical wave filters are becoming very important for use in noise
reduction and control of vibrations in all types of vibrating systems.
A number of books and numerous articles have been published on elec-
trical wave filters. Therefore it is important to establish the analogies
between electrical, mechanical and acoustical wave filters so that the
information on electrical wave filters may be used to solve filter prob-
lems in mechanical and acoustical systems. It is the purpose of this
chapter to illustrate and describe the different types of electrical, me-
chanical rectilineal, mechanical rotational and acoustical wave filters.
6.2. Types of Wave Filters
The response characteristics of wave filters are widely different. Themore frequently used types are designated as follows:
Low Pass Wave Filters
High Pass Wave Filters
Band Pass Wave Filters
Band Elimination Wave Filters
1 Campbell, G. A., Bell System Tech. Jour., Vol. I, No. 2, 1922.2 Stewart, G. W., Phys. Rev., Vol. 20, No. 6, p. 528, 1922.
92
RESPONSE CHARACTERISTICS OF WAVE FILTERS 93
A low pass wave filter is a system which passes currents, velocities,
angular velocities or volume currents of all frequencies from zero up to a
certain frequency termed the cutoff frequency /c and which bars cur-
rents, velocities, angular velocities and volume currents of all higher
frequencies.
A high pass wave filter is a system which passes currents, velocities,
angular velocities or volume currents of all frequencies from infinity
down to a certain frequency termed the cutoff frequency /c and which
bars currents, velocities, angular velocities and volume currents of all
lower frequencies.
A band pass wave filter is a system which passes currents, velocities,
angular velocities or volume currents that lie between two cutoff fre-
quencies /ci and/c2 and bars currents, velocities, angular velocities and
volume currents of all frequencies outside this range.
A band elimination wave filter is a system which bars currents, veloc-
ities, angular velocities or volume currents that lie between the two
cutoff frequencies Jex and Jc2 and passes currents, velocities, angular
velocities and volume currents of all frequencies outside this range.
6.3. Response Characteristics of Wave Filters ^' *
The ideal or non-dissipative filters consist entirely of pure reactances.
The primary object is the determination of those combinations of re-
actances which will give a single or double transmitted band of fre-
quencies. The most important type of structure is the ladder type,
that is, a certain combination of reactances in series with the line and
another combination in shunt with the line. The series reactance and
shunt reactance are designated by Zi and 22, respectively. It has been
shown in treatises on wave filters that attenuation occurs when 21/22 is
positive and when 21/22 is negative and no greater in absolute magnitude
than four. Non-attenuation occurs when 21/22 is negative and is less
in absolute magnitude than four. Therefore, a non-dissipative recur-
rent structure of the ladder type having series impedances 21 and shunt
impedances 22 will pass readily only currents of such frequencies as will
make the ratio 21/22 lie between and —4.
^ Johnson, "Transmission Circuits for Telephonic Communication," D. VanNostrand Co., New York.
*' Shea, "Transmission Networks and Wave Filters," D. Van Nostrand Co.,
New York.
94 WAVE FILTERS
6.4. Low Pass Wave Filters
Electrical, mechanical rectilineal, mechanical rotational and acous-
tical low pass filters are shown in Fig. 6.1.
RECTILINEAL
ELECTRICAL
<^A M Ca
- >
FREQUENCY
ACOUSTICAL
ROTATIONAL
MECHANICAL
Fig. 6.1. Electrical, mechanical rectilineal, mechanical rotational and acoustical low
pass wave filters.
The impedance of the series arm in the four systems is
zei = jo>L
zmi = ji^in
zri = i"/
ZAl = j(^M
The impedance of the shunt arm in the four systems is
1
ZE2
ZM2
ZR2
j'ojCe
1
JwCm
1
1
Z42 = Tj(x,Ca
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
HIGH PASS WAVE FILTERS 95
The limiting frequencies are given by
- = and - = -4 6.9
From the constants of the systems,
— = LCewc' = 0, when oip = 6.10ZE2
—— = mCMWc"^ = 0, when coe = 6.112A/2
— = /Cijojc^ = 0, when ccc = 6.122K2
— = MCicoc^ = 0, when coc = 6.13Z^2
— = — ZCeojc^ = — 4, when coc = , 6.142£2 VLCj'£
zm^i = — wCjfcoc^ = — 4, when coc = — . 6.152Af2 V WC,=V/
2iJl ,-^ 2= —ICrwc = — 4, when wc = —/ 6.16
—-^ = — MC4uc^ = — 4, when uc = , 6.172A2 VMC^
Equations 6.10 and 6.17, inclusive, show that the systems of Fig. 6.1
are low pass filters transmitting currents, linear velocities, angular veloc-
ities or volume currents of all frequencies lying between and the cutoff
frequency/c where/c = coc/27r.
6.5. High Pass Wave Filters
Electrical, mechanical rectilineal, mechanical rotational, and acousti-
cal high pass wave filters are shown in Fig. 6.2.
96 WAVE FILTERS
The impedance of the series arm in the four systems is
1
"ii'i
BAND PASS WAVE FILTERS 91
The limiting frequencies are given by
- = and - = -4 6.2622 22
From the constants of the systems,
6.272£1
98 WAVE FILTERS
The impedance of the series arm in the four systems is
1
1
zei = joiLi +
Zmi = jwmi +
Zri = jwli +
2^1 = jwMi +
1
1
jwQ
6.35
6.36
6.37
6.38Al
For a description of the acoustical capacitance Cai see Sec. 5.12.
BAND PASS WAVE FILTERS
2iJ21 — W Cb2-^2
Let
^^^1 - co^C^sAfa
The limiting frequencies are given by
?1 = and ^ = -4
LiCei = L2CE2
midii = rn2CM2
IiCri = I2CR2
MiCai = M2CA2
— =0, when wci = —
7
= —7=ZiS2 VLiCei VL2CJS2
= 0, when coci = , = /
— =0, when coci = , = ,
Z«2 V/iGji V/2Cfl2
ZAl „ ,1 1— = 0, when wc\ — —7== — —
/2.12 VMiCai VM2C42
Z£l ^(1 — WC2 LiCex) .— = — 4, when ., , _ = 4,
or
Z£2
'«>C2
ZAfl
2.V/2
a)c2 L2CE1
1 1
LiCe2 UCe\ \^LiCiE2
, (1 - WC2^WiC,lfi)^-4, when ^
—-; = 4,a)C2 m2tMi
99
6.41
6.42
6.43
6.44
6.45
6.46
6.47
6.48
6.49
6.50
6.51
6.52
100
or
WAVE FILTERS
'^C2 ±<xCm2 mxCux y/miCi.1/2
6.53
— = -4, when j-r~ = 4,ZiJ2 <^C2 -'2(>i21
or
i^C2 ~1 1 ]_
/iCb2 /iCri V/jCjR26.54
2.41, ,
(1 - ^C2'MiCaxY .— = -4, when --—; = 4,ZA2 <^C2 '^2(^Al
or
WC21 1
+MiCa2 MiCax VMiCa
6.55
It will be noted that uc2 has two values, one greater than uci and
one less than coci- Therefore, the upper and lower cutoff frequencies
are given by'
1
and
and
UC2 =
^C2
WC'2 =
WC2 =
Q, +1
LxCe2 LiCei 'VLiCe2
1U +1
+LiCe2 LiCei 'V LiCe2
1
' Tn^ f+
1
\Cm2 ntiCMi V WiC.v/2
11 1
+
C0C2
OTlCv2 WiCi/i •\/»ZiC;i/2
11 1
^iCb2 ^iCbi 'N//iCk2-
6.56
6.57
6.58
6.59
6.60
BAND ELIMINATION WAVE FILTERS 101
and
<^C2
0>C2 =
1
+1 1
I\Cr2 IiCiRl
1
+1
VhCB2
1
ind
(^02
MiCa2 MiCai VMiCa2
1
Vi^1
MiCa2 MiCai VMiCa•12
6.61
6.62
6.63
Equations 6.56 to 6.63, inclusive, show that the systems of Fig. 6.3
are band pass filters transmitting currents, linear velocities, angular
velocities or volume currents of all frequencies lying between two cutoff
frequencies /c2' and/c2", where /c2' = <^C2 I'^t^ and/c2" = coc2'72'r.
6.7. Band Elimination Wave Filters
Electrical, mechanical rectilineal, mechanical rotational and acous-
tical band elimination wave filters are shown in Fig. 6.4.
The impedance of the series arm in the four systems is
Zei
Za/i
Zffil
7coZi
CO CeiLi
jccmi
^CMimi
j^I\
1 ^^Cu\Ix
2.41 =JcoMi
1 - o?CaxMx
The impedance of the shunt arm in the four systems is
J2£2 = j'^Li
ZM2 = J(^fn2
UiCe2
J
uCm2
6.64
6.65
6.66
6.67
6.68
6.69
102 WAVE FILTERS
JZiJ2 = j<^l2
Za2 = JC0M2 —
6.70
6.71
BOTATrONAL
MECHANICAL
Fig. 6.4. Electrical, mechanical rectilineal, mechanical rotational and acoustical bandelimination wave filters.
The limiting frequencies are given by
Let
?i = and ^ = -422 22
L\Ce\ = L2CE2
miCMi = m^CMi
I\Cu\ = I2CR2
G.ll
6.73
6.74
6.75
6.76
BAND ELIMINATION WAVE FILTERS 103
— = 0, when wci = and coc4 = oo 6.77ZE2
—— = 0, when uci = and cca = oo 6.78ZM2
— = 0, when ojci = and 01^4 = 00 6.79ZR2
^— = 0, when oici = and wa = =0 6.80ZA2
Two of the Hmiting frequencies are determined by wci and wa above.
zei . ,LiCe2— = -4, when 2r n \Ze2 (1 — '^C ^li-El)
VLiCe2 + 16LiCei ± VLiCe2
Zmi .1 miCj
or
- -4, when- ""1^^^ = 4,ZM2 (1 — i^C miLMl)
VotiCi/2 + ]6wiCm ± 'VmiCM2 , or,ojc = T—7; 6.82
Zr\,
I\Cr2— = -4, when- 2r n \Zr2 (1 — WC -'iWnj
or
coc =4/1 Cj
6.83iji
Z^l . , MiCa2 ,— = -4, when^
= 4,2.42 (1 — UlC M\Ca\)
6.84
104 WAVE FILTERS
The other two limiting frequencies are
CdC2 = .
r ^ > D-O-"
and
tocs = o.oo
C0C2 = ^ ^ , O.O/
and
^cz =-.
7; o.t
WC3
CHAPTER VII
TRANSIENTS
7.1. Introduction
Transients embrace a wide variety of physical phenomena. An elec-
trical transient is the current which flows in a circuit following an elec-
trical disturbance in the system. A mechanical transient is the rectilinear
or angular velocity which occurs in a mechanism following a mechanical
disturbance in the system. An acoustical transient is the volume cur-
rent which flows in an acoustical system following an acoustical dis-
turbance in the system. The preceding sections have been concerned
with electrical, mechanical and acoustical systems in a steady state con-
dition. The formulas and expressions assume that the systems are in a
steady state condition of operation which means that the currents,
linear velocities, angular velocities or volume currents have become con-
stant direct or periodic functions of time. The steady state solution is
only one part of the solution because immediately after some change in
the system the currents or velocities have not settled into a steady state
condition. Electrical, mechanical and acoustical systems are subjected
to all types of varying and impulsive forces. Therefore, it is important
to examine the behavior of these systems when subjected to impulsive
forces as contrasted to steady state conditions.
The behavior of a vibrating system may be analyzed by solving the
difi^erential equations of the dynamical system. In other words find the
currents or velocities of the elements which when substituted in the
differential equations will satisfy the initial and final conditions. Thesolution of the differential equation may be divided ' into the steady
state term and the transient term. The operational calculus is of great
value in obtaining the transient response of an electrical, mechanical or
acoustical system to a suddenly impressed voltage, force or pressure.
^ Usually these parts are obtained by solving the differential equation for aparticular integral and a complementary function.
105
106 TRANSIENTS
The general analysis used by Heaviside is applicable to any type of
vibrating system whether electrical, mechanical or acoustical. Theresponse of a system to a unit force can be obtained with the Heaviside
calculus. It is the purpose of this section to determine the response of
electrical, mechanical rectilineal, mechanical rotational and acoustical
systems to a suddenly applied unit electromotive force, force, torque or
pressure respectively.
7.2. The Heaviside Operational Calculus ^^ '• ^
Heaviside's unextended problem is as follows: given a linear dynamical
system of « degrees of freedom in a state of equilibrium, find its response
when a unit force is applied at any point. The unit function, 1, depicted
in Fig. 7.1, is defined to be a force which is zero for / < and unity for
t> 0.
aT
t = o
Fig. 7.1. The unit function. The electromotive force, force, torque or pressure is zero
before and unity after / = 0.
The response of a dynamical system to a unit force is called the indicial
admittance of the system. It is denoted by A(t). A{t) represents the
current, linear velocity, angular velocity, or volume current when a unit
electromotive force, force, torque or pressure is suddenly applied in a
system which was initially at rest.
In the Heaviside calculus the differential equations are reduced to an
algebraic form by replacing the operator djdt by the operator p and the
operation / dt by 1/^. Tables of operational formulas have been com-
piled which serve for operational calculus the same purpose that tables
^ Carson, "Electric Circuit Theory and Operational Calculus," McGraw-HillBook Co., New York.
' Bush, "Operational Circuit Analysis," John Wiley and Sons, New York.*' Berg, "Heaviside's Operational Calculus," McGraw-Hill Book Co., New
York.
TRANSIENT RESPONSE 107
of integrals serve the integral calculus. Operational formulas may be
modified, divided or combined by various transformation schemes.
This is similar to integration by parts or change of variable in the
integral calculus.
The procedure in the Direct Heaviside Operational Method to be
followed in obtaining an operational solution of an ordinary differential
equation is as follows: Indicate differentiation with respect to the inde-
pendent variable by means of the operator p. Indicate integration by
means of \/p. Manipulate p algebraically and solve for the dependent
variable in terms of p. Interpret and evaluate the solution in terms of
known operators.
7.3. Transient Response of an Inductance and Electrical Resistance in
Series and the Mechanical Rectilineal, Mechanical Rotational and
Acoustical Analogies
The differential equation of an electromotive force, electrical resist-
ance and inductance connected in series, as shown in Fig. 7.2, is
di .
L -~ -{- tei = e 7.1at
where L = inductance, in abhenries,
r/j = electrical resistance, in abohms,
i = current, in abamperes, and
e = electromotive force, in abvolts.
Let the symbol p stand for the operator d/dt, then equation 7.1
becomes
Lpi -f rEi = e 7.2
The electrical admittance is
e te + Lp7.3
If ^ = for / < and unity for / > 0, then the ratio i/e is called the
electrical indicial admittance designated as Asit). The electrical indicial
admittance is
VE + Lp
108 TRANSIENTS
Equation 7.4 may be written
AE{i) =
where aE = Te/L.
1
(ajj + p)L
ELECTRICAL
PTaHM
RECTrLINCAL
I
ACOUSTICAL
7.5
ROTATIONAL
MECHANICAL
Fig. 7.2. Response of an electrical resistance and inductance in series and the analogous
mechanical rectilineal, mechanical rotational and acoustical systems to a unit electro-
motive force, unit force, unit torque and unit pressure, respectively. The graph depicts
the current, velocity, angular velocity or volume current as a function of the time for
unit excitation.
From the tables of operational formulas the solution of equation 7.5 is
1
^Eit) =^ (1 ") 7.6
Je(0 = ^ (1 - e-^-i')
Te1.1
The response characteristic is shown in Fig. 7.2. The current is zero
for / = 0. The current increases for values of / > and approaches the
value l/rg.
The differential equation of a force driving a mechanical rectilineal
resistance and mass, as shown in Fig. 7.2, is
dvm~ + tmV = jM 7.8at
where m = mass, in grams,
tm = mechanical rectilineal resistance, in mechanical ohms,
V = linear velocity, in centimeters per second, and
/m — driving force, in dynes.
TRANSIENT RESPONSE 109
The operational equation becomes
mpv + tmv = Jm 7.9
If/v/ = for / = and unity for / > then the ratio v/Jm is called
the mechanical rectilineal indicial admittance designated as AMif)-
The mechanical rectilineal indicial admittance is
AM{i)= ^1 7.10Tm + nip
Equation 7.10 may be written
^^^W=7 ^T^l 7.11[otM -t p)m
where au = ru/fn-
From the tables of operational formulas the solution is
^i/W = -^ (1 - e"-'") 7.12muM
or
J,j{t) = — (1 - e"^') 7.13
The response characteristic is shown in Fig. 7.2. The velocity is zero
for / = 0. The velocity increases for values of / > and approaches the
value 1/riv/.
The differential equation of a torque driving a mechanical rotational
resistance and moment of inertia, as shown in Fig. 7.2, is
deI^ + rRe=fR 7.14at
where / = moment of inertia, in gram (centimeter)^,
rj{ = mechanical rotational resistance, in rotational ohms,
d = angular velocity, in radians per second, and
Jr = torque, in dyne centimeters.
The operational equation becomes
Ipe + rne^jR 7.15
If//j = for / < and unity for t > Q then the ratio O/Jr is called the
mechanical rotational indicial admittance designated as AR(t).
110 TRANSIENTS
The mechanical rotational indicial admittance is
^^W=-4-77l 7.16Tr + Ip
Equation 7.16 may be written
^B{t)=-, ^—^rA 7.17""^{au + P)I
where aji — Vr/I.
From the tables of operational formulas, the solution of equation 7.17
is
^R{t) =~{l- 6-"') 7.18
or
^R{t) = -^ (1 - 6-T^') 7.19Tr
The response characteristic is shown in Fig. 7.2. The angular velocity
is zero for / = 0. The angular velocity increases for values of / > and
approaches the value l/rg.
The differential equation of a sound pressure driving an acoustical
resistance and inertance, as shown in Fig. 7.2, is
M^+rAU=p 7.20dt
where M = inertance, in grams per (centimeter)*,
ta = acoustical resistance, in acoustical ohms,
U = volume current, in cubic centimeters, and
p = sound pressure, in dynes per square centimeter.
The operational equation becomes
MpU+rAU = p 7.21
If ^ = for / < and unity for / > 0, then the ratio U/p is called
the acoustical indicial admittance designated as AA{t).
The acoustical indicial admittance is
Ji{t) = ^1 7.22^ ' TA + Mp
TRANSIENT RESPONSE 111
Equation 7.22 may be written,
{(XA + P)Mwhere a^ = rAJM.
From the tables of operational formulas, the solution of equation 7.23
is
AA{t) = - - (1 - 6--') 7.24MaA
or
JA{t) = " (1 - e-^ 7.25
The response characteristic is shown in Fig. 7.2. The volume current
is zero for t = 0. The volume current increases for values of / > and
approaches the value l/r^.
7.4. Transient Response of an Electrical Resistance and Electrical
Capacitance in Series and the Mechanical Rectilineal, Mechani-
cal Rotational and Acoustical Analogies
The differential equation of an electromotive force, electrical resistance
and electrical capacitance connected in series, as shown in Fig. 7.3, is
i /'e = rEi + -p7 I idt 1 .26
where Ce = electrical capacitance, in abfarads,
te = electrical resistance, in abohms,
/ = current, in abamperes, and
e = electromotive force, in abvolts.
The electrical indicial admittance is
^.w =y^^ 1 = r~,-r 1 7.271 + prE^E {oiE + P)rE
where «£• — XIteCe-
From the table of operational formulas, the solution of equation 7.27
IS t_-ocEt ^-r„CE
AE^t) = = 7.28rs ve
112 TRANSIENTS
The response characteristic is shown in Fig. 7.3. The current is l/rg
for / = 0. The current decreases for values of / > and approaches the
value zero as a limit.
ELECTRICAL
p =rA Ca :
1_ACOUSTICAL
ROTATIONAL
MECHANICAL
Fig. 7 3. Response of an electrical resistance and electrical capacitance in series and the
analogous mechanical rectilineal, mechanical rotational and acoustical systems to a unit
electromotive force, unit force, unit torque and unit pressure, respectively. The graph
depicts the current, velocity, angular velocity or volume current as a function of the
time for unit excitation.
The differential equation of a force driving a mechanical rectilineal
resistance and compliance, as shown in Fig. 7.3, is
1 rJm = tmV +^ \ 'odt
where Cm = compliance, in centimeters per dyne,
tm = mechanical rectilineal resistance, in mechanical ohms,
V = linear velocity, in centimeters, and
/m = force, in dynes.
The mechanical rectilineal admittance is
PCm ^ P ^
7.29
^M{t) = 1 =1 + PtmCm {oiM + P)rM
7.30
where a.y = l/rjfCv/.
From the tables of operational formulas, the solution of equation 7.30
Aiiit) =— ocMt ~
r<
7.31rM Tm
TRANSIENT RESPONSE 113
The response characteristic is shown in Fig. 7.3. The Hnear velocity
is 1/rif for t = 0. The velocity decreases for values of / > and ap-
proaches the value zero as a limit.
The differential equation of a torque driving a mechanical rotational
resistance and rotational compliance, as shown in Fig. 7.3, is
fR = rBd+— I edt 7.32Cij J
where Cr = rotational compliance, in radians per dyne per centimeter,
vr = mechanical rotational resistance, in rotational ohms,
= angular velocity, in radians per second, and
fu = torque, in dyne centimeters.
The mechanical rotational indicial admittance is
^.W=T^^1 =—fr^l 7.331 + prRtR {uR + p)rR
where aR = IIvrCr.
From the tables of operational formulas the solution of equation 7.33
is
AR{i) = = 7.34Tr Vr
The response characteristic is shown in Fig. 7.3. The angular velocity
is 1/rB for / = 0. The angular velocity decreases for values of / >and approaches the value zero as a limit.
The differential equation of a sound pressure driving an acoustical
resistance and an acoustical capacitance, as shown in Fig. 7.3, is
p = rAU+^ fudt 1.2S
where Ca = acoustical capacitance, in (centimeter)^ per dyne,
Ta = acoustical resistance, in acoustical ohms,
U = volume current, in cubic centimeters per second, and
p = sound pressure, in dynes per square centimeter.
114 TRANSIENTS
The acoustical indicial admittance is
A,{t) = ^^^1 = ,
^,
1 7.361 + PvaCa (a + Pfa
where aa = ^ItaCa-
From the tables of operational formulas the solution of equation 7.36
is(
AA{t) = = 7.37TA TA
The response characteristic is shown in Fig. 7.3. The volume current
is l/r^i for / = 0. The volume current decreases for values of / > and
approaches the value zero as a limit.
7.5. Transient Response of an Electrical Resistance, Inductance andElectrical Capacitance in Series and the Mechanical Rectilineal,
Mechanical Rotational and Acoustical Analogies
The differential equation of an electromotive force, electrical resist-
ance, inductance and electrical capacitance connected in series, as shown
in Fig. 7.4, is ^.J
.
L-j + rsi^--: / idt = e 7.38dt Ce J
where L = inductance, in abhenries,
rs = electrical resistance, in abohms,
Ce = electrical capacitance, in abohms,
/ = current, in abamperes, and
e = electromotive force, in abvolts.
The electrical indicial admittance is
4eW = 1 7.39
Lp^ + rEp + ~Let
TE
= ^77^J^f-'E
TRANSIENT RESPONSE
The electrical indicial admittance is
115
7.40LwE ip + as)^ + co£^
From the tables of operational formulas, the solution of equation 7.40
4eW = 7— e"""' sin w^tL03E
7.41
: fELECTRICAL ^
ROTATIONAL
MECHANICAL
Fig. 7.4. Response of an electrical resistance, inductance and electrical capacitance in
series and the analogous mechanical rectihneal, mechanical rotational and acoustical
systems to a unit electromotive force, unit force, unit torque or unit pressure, respec-
tively. The graph depicts the current, velocity, angular velocity or volume current as a
function of the time for unit excitation.
The response for te^ < AL/Ce is shown in Fig. 7.4.^. It is a dampedsinusoid.
If Te^ > AL/Ce, then the solution becomes
^E{t) = —r e-"^' sinh fe/
where fe = \/<^£= V"^' LCe
7.42
The response for this condition is shown in Fig. 7.45.
116 TRANSIENTS
If te^ = AL/Ce, then sin cast approaches cob/ and the solution is
Jj^{t) = - e"""^' 7.43
The response for this condition is shown in Fig. 1AC.
The differential equation of a force driving a mass, mechanical recti-
lineal resistance and compliance is shown in Fig. 7.4 as
Xmx + tmX + -p;- = Jm 1A\
Cmwhere m = mass, in grams,
rM = mechanical rectilineal resistance, in mechanical ohms.
Cm = compliance, in centimeters per dyne,
X = acceleration, in centimeters per second per second,
X — velocity, in centimeters per second,
X = displacement, in centimeters, and
Jm = driving force, in dynes.
Substituting v for x, equation 7.44 may be written
7 1 /"
m-^ + tmv + 7:r / ^^^ — Jm '7-45
'M Jdt C
The mechanical rectilineal indicial admittance is
AMit) = 1 7.46
mp^ + tmP + -pr^M
Let
tmO-M =
m
'' mLM
The mechanical rectilineal indicial admittance is
^-« = -^r. ^ ^7^ 2
1 7.47mwM {P + oim) + ^M
TRANSIENT RESPONSE 117
From the tables of operational formulas the solution is
Auii) = £-""' sin mit 7.48
The response for r^^ < '^m/Cu is shown in Fig. lAA. It is a damped
sinusoid.
If tm^ > '^in/Cu, then the solution becomes
Am {t) € ~ ""' sinh fe/ 7.49
M = Vfe'where ftv/ = \ B^f^mCiM
The response of this condition is shown in Fig. 7.4_S.
If rM^ — Am/Cm then sin oimI approaches oiMt and the solution is
J^[t) = - e-""" 7.50m
The response for this condition is shown in Fig. 7.4C
The differential equations of a torque driving a moment of inertia,
mechanical rotational resistance and rotational compliance, as shown in
Fig. 7.4, is
/0 + rij0 + ~ =/ij 7.51
where / = moment of inertia, in grams (centimeter),
vr = mechanical rotational resistance, in rotational ohms,
Cr = rotational compliance, in radians per dyne per centimeter,
4> = angular acceleration, in radians per second per second,
<J)= angular velocity, in radians per second,
4> = angular displacement, in radians, and
Jr = driving torque, in dyne centimeters.
Substituting 6 for 4>, equation 7.51 may be written
de 1
7, + ""' + &'^^ + rRd^~ fedt=fR l.si
'R J
118 TRANSIENTS
The mechanical rotational indicial admittance is
ARif) = ^J-
1 7.53
Let
traR =
j
-IThe mechanical rotational indicial admittance is
Ii^R (P + cxr)^ + WR^^i^» = -V- ,. , \2^ 2 1 7.54
From the tables of operational formulas the solution is
AR{i) = -r- €-"•'' sin coij/ 7.55l^R
The response for tr"^ < ^I/Cr is shown in Fig. lAA. It is a dampedsinusoid.
If rR^ > AI/Cr, then the solution becomes
^ieW =7^ «"'""sinh/3/e^ 7.56^PR
where ^^j = Jo,^2 _
The response for this condition is shown in Fig. 7.45.
If rR^ = '^I/Cr, then sin wrI approaches dORt and the solution is
AR{t) =Je--' 7.57
The response for this condition is shown in Fig. 7.4C.
TRANSIENT RESPONSE 119
The differential equation of a sound pressure driving an acoustical
resistance and an inertance connected to an acoustical capacitance, as
shown in Fig. 7.4, is
MX + rAX\~ = p 7.58
where M = inertance, in grams per (centimeter)^,
r^ = acoustical resistance, in acoustical ohms,
Ca = acoustical capacitance, in (centimeter)^ per dyne,
X = volume current, in cubic centimeters per second, and
p — pressure, in dynes per square centimeter.
Substituting U for X, equation 7.58 may be written
if'M— +rAU +~ I Udt = p 7.59at La
The acoustical indicial admittance is
^A{t) = rl 7.60
Let
OCA
The acoustical indicial admittance is
Mp^
120 TRANSIENTS
IfrA^ > 4M/Ca, then the solution becomes
^a(()1
Li3a
' sinh 0Af 7.63
where /3a = -YaA MCa
The response for this condition is shown in Fig. 7.45.
If r4^ = 4M/Ca, then sin oj^/ approaches cca^ and the solution is
^A{t) = ^ — acAt
M IM
The response for this condition is shown in Fig. 7.4C
7.6. Arbitrary Force
In the preceding sections the response of various combinations of
elements to a unit force has been obtained. The value of the unit force
t=o
Fig. 7.5. Step function approximation.
solution is that the response to any arbitrary force can be obtained from
the unit force solution by a single integration of Duhamel's integral.
ARBITRARY FORCE 121
It is the purpose of this section to illustrate the proof and use of this
integral.
The discussion will be confined to the electrical system. This proof,
as in the case of the preceding sections, can be extended to apply to
mechanical and acoustical systems. Let the arbitrary electromotive
force be represented by the curve of Fig. 7.5. The curve can be assumed
to be made up of a series of unit type electromotive forces, as shown in
Fig. 7.5. At / = an electromotive force eg is impressed upon the sys-
tem. A time Ui later, an electromotive force ei is added, a time «2 later,
an electromotive force ^2 is added, etc., all being of the unit type. Thecurrent at a time / is then the sum of the currents due to e^ at i = 0,
<?i at / = «i, etc. The current due to eo is ^q-^^Wj where ^eU) is the
indicial electrical admittance. The current due to the electromotive
force Ae, which begins at time «, is obviously AeAsit — u), t — u
being the time elapsed since the unit electromotive force Ae was turned
on. Therefore, the total current at the time « = / is
u = t
i = eoAEit) +^^AeAE{t - u) 7.65
d^ = — e{u)du
du /t J
AE{t — u) -y e{u)du 7.66du
The above expression may be transformed into different forms.
The integral may be transformed by integrating as follows:
/ UdF = Uv\- f VdU 7.67Jii Jo Jo
U = A{t- u)
dV = de{ii)
V = e{u)
Making the above substitutions, a new expression for the current
follows
:
e{u) — 4e(/ - u)du 7.68du
122 TRANSIENTS
Equation 7.68 is a fundamental formula which shows the mathe-
matical relation between the current and the type of electromotive
force and the constants of the system.
The important conclusions regarding Duhamel's integral are as fol-
lows: The indicial admittance of an electrical network determines within
a single integration the behavior of a network to any type of electro-
motive force. In other words a knowledge of the indicial admittance is
the only information necessary to completely predict the performance
of a system including the steady state.
The velocity, angular velocity and volume current in the mechanical
rectilineal, mechanical rotational and acoustical systems are analogous
to the equation for the current in the electrical system. Therefore
Duhamel's integrals in the mechanical rectilineal, mechanical rotational
and acoustical systems are as follows:
V =fM{t)AM{(y) + / fM{u) — Auit - u)du 7.69
fR{u) — AR{t - u)du 7.70du
p{u)— AA{t - u)du in\du
The following general conclusion can be stated as follows: The indicial
admittance of any vibrating system determines within a single integra-
tion the behavior of the system to any type of applied force.
The Heaviside calculus then becomes an important tool in the solu-
tion of transient problems in mechanical and acoustical systems. Since
a great number of problems in these fields involve impulsive forces the
use of analogies makes it possible to use the tremendous storehouse of
information on electrical systems for the solution of problems in me-
chanical and acoustical systems.
As an example illustrating the use of Duhamel's integral consider an
electromotive force Ee"^* impressed on the electrical circuit of Fig. 7.3
consisting of an electrical resistance and electrical capacitance in series.
The indicial electrical admittance from equation 7.28 is
4g(/) = — e"c^' 7.72
CHAPTER VIII
DRIVING SYSTEMS
8.1. Introduction
An electromechanical or electroacoustic transducer or driving system is
a system for converting electrical vibrations into the corresponding me-chanical or acoustical vibrations. The most common driving systems in
use to-day for converting electrical variations into mechanical vibrations
are the electrodynamic, the electromagnetic, the electrostatic, the mag-netostrictive and the piezoelectric. It is the purpose of this chapter to
consider the electrical and mechanical characteristics of these driving
systems.
8.2. Electrodynamic Driving System
A moving coil or dynamic driving system is a driving system in
which the mechanical forces are developed by the interaction of currents
CROSS-SECTIONAL VIEW
[-VW—/OOJTH
ZCN
ELECTRODYNAMIC DRIVING SYSTEM 125
in a conductor and the magnetic field in which it is located. The system
is depicted in Fig. 8.1. The force, in dynes, due to the interaction of
the current in the voice coil and the polarizing field is
Jm = Bli 8.1
where B = flux density, in gausses,
/ = length of the conductor, in centimeters, and
/ = current, in abamperes.
The electromotive force, in abvolts, developed by the motion of the
conductor is
e = BIx 8.2
where x = velocity, in centimeters per second.
From equations 8.1 and 8.2
- = (S/)2 4- 8-3i jM
- = zem 8.4
where Zem = electrical impedance, in abohms, due to motion, termed
motional electrical impedance.
From the mechanical circuit ' of Fig. 8.1, the mechanical rectilineal
impedance of the vibrating system at the voice coil is
zm = zmi + zm2 8.5
' In the illustrations in the preceding chapters the elements in the electrical
network have been labeled rE, L and Ce- However, in using analogies in actual
practice, the conventional procedure is to label the elements in the analogous
electrical network with ru, m and Cm for a mechanical rectilineal system,
with vr, I and Cr for a mechanical rotational system and with va-, M and Cafor an acoustical system. This procedure will be followed in this chapter in
labeling the elements of the analogous electrical circuit. It is customary to
label this network with the caption "analogous electrical network of the mechani-cal system" (or of the rotational system or of the acoustical system) or with the
caption "mechanical network" (or rotational network or acoustical network).
The latter convention will be used in this chapter. When there is only one path,
circuit will be used instead of network.
126 DRIVING SYSTEMS
where zm = the total mechanical rectilineal impedance at the conduc-
tor, in mechanical ohms,
ZjVfi = the mechanical rectilineal impedance of the voice coil and
suspension system, in mechanical ohms, and
Zm2 — the mechanical rectilineal impedance of the load, in me-
chanical ohms.
The mechanical rectilineal impedance at the voice coil is
ZM = — 8.6X
The electrical impedance due to motion from equations 8.3, 8.4 and
8.6 is
zem = 8.7Zm
The motional electrical impedance of a transducer is the vector differ-
ence between its normal and blocked electrical impedance.
The normal electrical impedance of a transducer is the electrical im-
pedance measured at the input to the transducer when the output is con-
nected to its normal load.
The blocked electrical impedance of a transducer is the electrical im-
pedance measured at the input when the mechanical rectilineal system is
blocked, that is, in the absence of motion.
The normal electrical impedance zek, in abohms, of the voice coil is
Zen = zbi + Zem 8.8
where z^i = damped electrical impedance of the voice coil, in abohms,
and
Zem = motional impedance of the voice coil, in abohms.
The motional electrical impedance as given by equation 8.8 may be
represented as in series with the blocked or damped electrical impedance
of the conductor, as depicted by the electrical circuit in Fig. 8.1.
The dynamic driving system is almost universally used for all types of
direct radiator and horn loud speakers.
8.3. Electromagnetic Driving Systems
A magnetic driving system is a driving system in which the mechanical
forces result from magnetic reactions. There are three general types of
ELECTROMAGNETIC DRIVING SYSTEMS 127
magnetic driving systems, namely, the unpolarized armature type, the
polarized reed type and the polarized balanced armature type.
A. Unpolarized Armature Type.—The unpolarized armature driving
system consists of an electromagnet operating directly upon an armature.
The armature is spaced at a small distance from the pole piece woundwith insulated wire carrying the alternating current. Since there is no
polarizing flux, the driving force frequency is twice the frequency of the
impressed current to the coil.
Consider the system shown in Fig. 8.2. Assume that all the reluctance
128 DRIVING SYSTEMS
Assume that the armature is displaced from its normal position a dis-
tance A«' centimeters towards the pole; the total flux is
Ci4>T + ^(t>T
= 8.10a — £iX
Now let the armature be displaced a distance Aa; centimeters awayfrom the pole; the total flux is
Ci<f>T
— A(f>T = 8.11a -\- Ax
2^"^^ = -^-77:X2 8.12
8.13
The difference in flux for these two conditions is
2C/A;v
a^ — (Ax)
The change in flux with respect to time is
A(f>T Ci Ax
At a^ — {Ax)^ At
The voltage, in abvolts, generated due to motion is
e = n— 8.14dt
\i Ax is small compared to a, then from equations 8.13 and 8.14
^Cie = ^r X 8.15
The force on the armature, in dynes, is
where / = current in the coil, in abamperes,
C = l-^nA,
A = area of the center pole, in square centimeters,
n = number of turns, and
a = normal spacing, in centimeters.
ELECTROMAGNETIC DRIVING SYSTEMS 129
If the current in the coil is sinusoidal, then the expression for the
current can be written
i = «max sin OJ/ 8.17
where /max = amplitude of the current in abamperes,
CO = 2irA
/ = frequency in cycles per second, and
/ = time, in seconds.
Substituting equation 8.17 for the current in 8.16, the force on the
armature is
Im = '. 71 «'^max Sin^ CO/
4iryfa
(|-|cos2co/) 8.18A ^ 2 * max '
Equation 8.18 shows that there is a steady force and an alternating
driving force of twice the frequency of the impressed current.
From the mechanical circuit of Fig. 8.2, the mechanical rectilineal
impedance of the vibrating system is
Z.V/ = 2a/ 1 + Zm2 8.19
where Zm = total mechanical rectilineal impedance at the armature, in
mechanical ohms,
Zmi = mechanical rectilineal impedance of the armature, in me-
chanical ohms, and
Zm2 = mechanical rectilineal impedance of the load, in mechanical
ohms.
The mechanical rectilineal impedance at the armature is
/mZm = — 8.20
X
From equations 8.15 and 8.16
e nC^P X
. ^ 4 . 8.211 4TrJa*/M
130 DRIVING SYSTEMS
From equations 8.19, 8.20 and 8.21
27r2«4^2-2zem =
i 8.22a zm
where zem = motional electrical impedance, in abohms,
n = number of turns,
^ = area of center pole, in square centimeters,
/ = current, in abamperes,
a = spacing, in centimeters, and
Zm = mechanical rectilineal impedance of the load including
the armature, in mechanical ohms.
The normal electrical impedance Zen, in abohms, of the coil is
Zen = zbi + Zem 8.23
where z^i — damped electrical impedance of the coil, in abohms, and
Zem = motional electrical impedance of the coil, in abohms.
The motional electrical impedance as given by equation 8.23 may be
represented as in series with the blocked or damped electrical impedance
of the coil as depicted by the electrical circuit in Fig. 8.2.
The frequency of vibration of the armature is twice the frequency of
the impressed electrical current. Therefore, this system cannot be used
for the reproduction of sound. It is, however, a simple driving system
for converting electrical variations into mechanical vibrations of double
frequency. The unpolarized driving system is used for low frequency
supersonic generators, saws, filing machines, vibrators and clippers.
B. Polarized Reed Armature Type.—A reed armature driving system
consists of an electromagnet operating directly upon an armature of steel
as in Fig. 8.3. The steel armature is spaced at a small distance from a
pole piece wound with insulated wire carrying the alternating current
and supplied with steady flux from the poles of a permanent magnet.
The flux, in maxwells, due to the permanent magnet is given by
M<^i = ^ 8.24
where M — magnetomotive force of the magnet, in gilberts, and
i?i = reluctance of the magnetic circuit, in oersteds.
ELECTROMAGNETIC DRIVING SYSTEMS 131
The flux, in maxwells, due to the sinusoidal current /max sin oit in the
coils is given by
47rM„,axsin wt<P2
=n o-^-i
where N — number of turns in the coil,
/ = current in the coil, in abamperes,
/?2 = reluctance of the alternating magnetic circuit, in oersteds,
CO = 27r/,
/ = frequency, and
t = time.
r ^^' L
COIL
aPOLE
Zmi
ARMATURE
MAGNET
r^M WS^
1 1
132 DRIVING SYSTEMS
quency as the alternating current and the last term represents a force of
twice the frequency of the alternating current. Referring to equation
8.26 it will be seen that the driving force is proportional to the steady
flux 4>i- Also <t>i must be large compared to (^2) in order to reduce second
harmonic distortion. For these reasons the polarizing flux should be
made as large as possible.
The motional electrical impedance of this system will now be con-
sidered. If all the reluctance is assumed to reside in the air gap, the flux,
in maxwells, through the armature is
4>i= 8.27
a
where M = magnetomotive force, in gilberts, due to the steady field,
a = spacing between the armature and pole, in centimeters, and
A = area of the pole, in square centimeters.
Let the armature be deflected a distance A^ towards the pole; the flux
will now be
MA4>x + A.^1 = 8.28
fl — A*
Now let the armature be pulled away from the normal position a
distance of A;^ ; the flux will be
MA4>i - Act>i = -—-— 8.29
a -\- Ax
The difference in flux through the armature for these two conditions is
MA MA __ IMAAx ^ IMAAxAx a -\- Ax a^ — (Ax)^ a
This change in flux with respect to the time is
d<t,i MAdxat a at
The electromotive force, in abvolts, generated in the coil due to this
deflection of the armature is
^(^1 NMA „ ^^e = A^-^ = ——X 8.32
at a
2A4>i = 7" -, ., .
= -^ y—r2 = 5— 8.30
ELECTROMAGNE'nC DRIVING SYSTEMS 133
Leaving out the steady force and the force of twice the frequency,
equation 8.26 becomes
MNifu = ~r—;;
—
~ 8.33
From the mechanical circuit of Fig. 8.3, the mechanical rectilineal
impedance of the vibrating system is
Zj>f = 2Afl + 2jW2 8.34
where zm = total mechanical rectilineal impedance at the armature
directly above the pole piece, in mechanical ohms,
Zjv/i = mechanical rectilineal impedance of the armature, in me-
chanical ohms, and
ZAf2 = mechanical rectilineal impedance of the load in mechanical
ohms.
The mechanical rectilineal impedances zm, ZAfi, and zmi are referred to
a point on the armature directly over the pole piece.
The mechanical rectilineal impedance of the armature directly above
the pole piece is
Zm = — 8.35X
Combining equations 8.32 and 8.33,
e _ _x_ M^N^i Jm R\R2a^
From equations 8.34, 8.35 and 8.36
8.36
ZeM — TT^—2°-37
RiR2a Zj^f
where z^m = motional impedance, in abohms,
Zm = total mechanical impedance with reference to a point on
the armature directly over the pole piece.
From equations 8.24 and 8.37, assuming Ri = i?2
zem — ""o 8.38a Zm
134 DRIVING SYSTEMS
Equation 8.38 is similar to equation 8.7 for the electrodynamic sys-
tem. The normal electrical impedance Zen, in abohms, of the coil is
Zen ~ zei -\- zem 8.39
where zei = damped electrical impedance of the coil, in abohms, and
Zem — motional electrical impedance of the coil, in abohms.
The motional electrical impedance as given by equation 8.38 may be
represented as in series with the blocked or damped electrical impedance
of the coil as depicted by the electrical circuit in Fig. 8.3.
This driving system is not generally used in loud speakers. The most
common example of this driving system is the bipolar telephone receiver
where the diaphragm is the armature.
C. Polarized Balanced Armature Type.—There are innumerable pos-
sibilities in the design of a magnetic driving system. The preceding
|-^vM
—
WS^
ME.ARMATURE
MAGNET
Zen
ELECTROMAGNETIC DRIVING SYSTEMS 135
A typical balanced armature driving system is shown in Fig. 8.4.
The steady field is usually supplied by a permanent magnet. The arma-
ture is located so that it is in the equilibrium with the steady forces. Thealternating current winding is wound around the armature. The steady
force, in dynes, at the poles (Fig. 8.4) due to the magnetic field is
-^- = ib '''
where 4>i = total flux, in maxwells, at each pole due to the permanent
magnet, and
yf = effective area, in square centimeters, of the pole piece.
The flux, in maxwells, at the poles due to a current in the coil is
4TrA^/
02 = -^- 8.41
where N = number of turns in the coil,
i = current in the coil, in abamperes, and
i?2 = reluctance of the magnetic circuit, in oersteds, which the
coil energizes.
The sum of the forces, in dynes, at the four poles acting upon the
armature due to a current in the coil is
, 2((^i + <^2) 2(<^i — <t>2) <j)l4>2
'%'wA i'wA tA
or
^- = -^ '''
In the case of the simple reed driving system a second harmonic
term appeared in the force when a sinusoidal current was passed through
the coil. It is interesting to note that in the case of the balanced arma-
ture the second harmonic term cancels out due to the push pull arrange-
ment.
The motional impedance of this system will now be considered. Let
the armature be deflected clockwise a distance of Ax from the poles. The
136 DRIVING SYSTEMS
flux, in maxwells, through the armature to the right and upward, assum-
ing that the entire reluctance exists in the air gap, is
MA<^i + A01 = 8.44
l(a — A;v)
where M = magnetomotive force, in gilberts, of the steady field,
a = spacing between the armature and pole, in centimeters, and
A = effective area of a pole piece, in square centimeters.
The flux through the armature to the left and downward is
MA2(a + DiX)
The flux through the armature is the difference between equations
8.44 and 8.45.
MA^x MAl^x
or — {Axy a^
The change in flux with respect to the time is
d<^ M dx MA— = -x— = —^ X 8.47at a at a
The electromotive force, in abvolts, generated in the coil is
^di, NMAe = N-i^ = —^x 8.48
dt a^
From the mechanical circuit of Fig. 8.4, the mechanical rectilineal
impedance of the vibrating system is
ZM = ZMl + Zm2 8.49
where Zm = total mechanical rectilineal impedance, in mechanical ohms,
zmi — mechanical rectilineal impedance of the armature, in me-
chanical ohms, and
zm2 — mechanical rectilineal impedance of the load, in mechanical
ohms.
ELECTROMAGNETIC DRIVING SYSTEMS 137
The mechanical rectilineal Impedances Zm, Zm\, and Zm2 are referred
to a point on the armature directly above a pole piece.
The mechanical rectilineal impedance at the armature directly over a
pole piece is
ZM =-^ 8.50
Combining equations 8.43 and 8.50,
138 DRIVING SYSTEMS
the large magnetic forces, the stiffness of the centering system must be
relatively large.
This driving system is used for loud speakers, galvanometers, for
motion picture film recording and for facsimile printers.
In actual practice it appears very difficult to reduce the stiffness
sufficiently so that the resonance of the system will occur below 100
cycles. Therefore, when this driving system is used for a loud speaker
the response will fall off quite rapidly below the resonance frequency.
8.4. Electrostatic Driving System
An electrostatic driving system is a driving system in which the
mechanical forces result from electrostatic reactions. Consider the
system of Fig. 8.5 consisting of a vibrating surface moving normal to
STATIONARYPLATE
DIAPHRAGM
^ Ze, c'em
1
ELECTROSTATIC DRIVING SYSTEM 139
Assume that the polarizing voltage is ^'o and that the alternating
voltage is f = frnax sin wt. The force, in dynes, between the plates is
, ((?o + ^max sin wifAJM = '
2 8-5oera
<?0^ + 2(?ofmax sin CO/ + i^^max — 2^^max COS 2wtJM = '—
2 A 8.57OTra
The first and third terms in the numerator of equation 8.57 represent
Steady forces. The fourth term is an alternating force of twice the fre-
quency of the impressed voltage. The second term is an alternating
force of the frequency of the impressed voltage. If the polarizing electro-
motive Cq is large compared to the alternating electromotive force d^max
sin oj/, the fourth term will be negligible. The useful force, in dynes,
then is the second term which causes the moving surface to vibrate
with a velocity which corresponds to the impressed electromotive force.
, ^o^max sin wt . eoe
The motional impedance of this system will now be considered.
The charge, in statcoulombs, on the condenser is
q = CeCq 8.59
where e^ = potential difference between the plates, in statvolts, and
Ce = capacity per unit area, in statfarads.
The current, in statamperes, generated due to motion is
I = —dt
From equations 8.59 and 8.60 the generated current is
8.60
dCE dx
dx dt
The capacitance of the condenser, in statfarads, is
ACei = —
-
8.624Tra
140 DRIVING SYSTEMS
Let the movable plate be deflected a distance h.x away from the fixed
plate. The capacitance is
/j
Cei - ACei =,
, , ,8.63
4ir(<2 + Ax)
Now let the movable plate be deflected a distance Ax towards the
fixed plate. The capacitance is
Cei + ACei = y-, 8.6447r(a — Ax)
The difference between the two conditions is
JAx ^ y4Ax
4w[a^ - (Axf]^
47r«'
The change in capacitance with respect to x is
dCEi 1^^
(ix 47r«^
^^^•1 = T^^T-77:72T ^ 7-2 8.65
8.66
Substituting equation 8.66 in 8.61, the generated current, in stat-
amperes, is
1 = -—5 X 8.67
From the mechanical circuit of Fig. 8.5, the mechanical rectilineal
impedance of the vibrating system is
zm = zmi + ZM'i 8.68
where Zm = total mechanical rectilineal impedance of the vibrating
system, in mechanical ohms,
Zmi = mechanical rectilineal impedance of the vibrating plate, in
mechanical ohms, and
Zm2 = mechanical rectilineal impedance of the load, in mechanical
ohms.
The mechanical rectilineal impedance at the plate is
ZM = — 8.69X
MAGNETOSTRICTION DRIVING SYSTEM 141
From equations 8.58 and 8.67
t e^A^ X
From equations 8.68, 8.69 and 8.70
167rV ^ _^
eo A
where Zeu = motional electrical impedance, in statohms, and
zm = total mechanical rectilineal impedance presented to the
vibrating surface including the vibrating surface.
The normal electrical impedance zeN} in statohms, of the condenser is
ZeiZeM „ -,Zen =
, 8.72ZjSI + zem
where zei = damped electrical impedance of the condenser. In statohms,
and
Zem = motional electrical impedance of the condenser, in stat-
ohms.
The motional electrical impedance as given by equation 8.72 may be
represented as in parallel with the blocked or damped electrical im-
pedance of the condenser as depicted by the electrical network in Fig.
8.5.
The condenser driving system has been employed as a loud speaker
in which case the moving electrode radiates directly into the air. Meansmust be provided to keep the electrodes separated without, at the same
time, adding a large stiffness. In a bilateral or push pull arrangement
the movable electrode is placed between two stationary plates and the
large steady forces are balanced out.
8.5. Magnetostriction Driving System
A magnetostriction driving system is a driving system in which the
mechanical forces result from the deformation of a ferromagnetic mate-
142 DRIVING SYSTEMS
rial having magnetostriction properties. The term "Joule effect" is
applied to the phenomena in which a change in linear dimensions occurs
when a magnetic field is applied along a specified direction. The term
"Villari effect" is applied to the phenomena in which a change in mag-
netic induction occurs when a mechanical stress is applied along a
specified direction.
Consider the system shown in Fig. 8.6. Assume that the rod is
clamped so that no motion is possible and that a. current is applied to
2m
MAGNETOSTRICTION DRIVING SYSTEM 143
The relation between flux and the current is
B = -—
-
8.74RJ
where N = number of turns,
/ = current in the coil, in abamperes,
R = reluctance of the magnetic circuit, and
/i = area of the rod, in square centimeters.
Combining equations 8.73 and 8.74 and ehminating the steady force,
AwNiK .
/m = —-— sm CO/ 8.75R
where K = constant representing the dynamical Joule magnetostriction
effect,
R = reluctance of the magnetic circuit,
N = number of turns in the coil,
/ = current, in abamperes,
w = Iwf,
f = frequency, in cycles per second, and
t = time, in seconds.
If the rod is allowed to vibrate, this stress may be considered to be
the driving force.
The electromotive force, in abvolts, induced in the winding, due to
the Villari effect, is
e = MJ^ 8.76dt
where A'^ = number of turns,
A = cross-sectional area of the nickel rod, in square centimeters,
and
B = magnetic induction, in gausses.
The magnetic induction is
„ 47rifB = -—- X 8.77
JR
where x = total extension of deformation, in centimeters, and
K = constant representing the dynamical Villari magnetostric-
tion effect.
144 DRIVING SYSTEMS
The induced voltage, in abvolts, is
From equation 8.75
e = —-— X 8.78K
JmR^tNK
In the above consideration it has been assumed that the stress and
driving force are uniform over the length of the rod. Under these condi-
tions the rod is a compliance given by
Cmi = -^ 8.80EA
where A = cross-sectional area of the rod, in square centimeters,
/ = length of the rod, in centimeters, and
E = Young's modulus.
The mechanical rectilineal impedance of the rod is
zmi = -^-^ 8.81JUiLmi
For the conditions under consideration the mechanical rectilineal im-
pedance of the vibrating system, from the mechanical circuit of Fig.
8.6, is
zm = ZAfi + zif2 8.82
where Zm = total mechanical rectilineal impedance, in mechanical ohms,
zmi = mechanical rectilineal impedance of the rod, in mechanical
ohms, and
Zm2 = mechanical rectilineal impedance of the load, in mechanical
ohms.
The mechanical rectilineal impedances zm, za/i and zm2 ^fe referred
to one end of the rod with the other end rigidly fixed. The dimensions
of the rod are assumed to be small compared to the wavelength.
MAGNETOSTRICTION DRIVING SYSTEM 145
The mechanical rectilineal impedance at the end of the rod is
ZM = — 8.83X
From equations 8.79, 8.82 and 8.83
zem = —r^ 8.84K Zm
where Zem = motional electrical impedance, in abohms, and
Zm = total mechanical rectilineal impedance load upon the rod,
including the effective mechanical rectilineal impedance
of the rod, in mechanical ohms.
The normal impedance of the coil is
Zen = Zfii + Zem 8.85
where z^i = damped impedance of the voice coil, in abohms, and
Zem = motional impedance, in abohms—equation 8.84.
The damped impedance of the coil of most magnetostriction systems
comprises a resistance in series with an inductance (Fig. 8.6). Thedamped impedance and the motional impedance are effectively in series,
as shown by equation 8.85 and depicted by the electrical circuit in
Fig. 8.6.
In the above considerations the length of the rod is assumed to be a
small fraction of the wavelength. In general, magnetostriction driving
systems ^ are operated at resonance. The three most common systems
are as follows: a rod fixed on one end and loaded on the other, a rod
free on one end and loaded on the other and a free rod. The lumped
constant representations of the three systems depicted by the mechanical
networks in Fig. 8.7 are valid near the resonant frequency of the rod.
The mass mi in Fig. 8.7 is given by
plJmi = — 8.86
^ Mason, "Electromechanical Transducers and Wave Filters," D. Van Nos-trand Co., New York, 1942.
146 DRIVING SYSTEMS
where p = density of the rod material, in grams per cubic centimeter,
/ = length of the rod, in centimeters, and
A = cross section of the rod, in square centimeters.
The compliance Cmi, in Fig. 8.7, is given by
Cmi = 2 7-^ 8.87
where A = cross-sectional area of the rod, in square centimeters,
/ = length of the rod, in centimeters, and
E = Young's modulus.
The compliance given by equation 8.87 is 8/7r^ times the static compli-
ance given by equation 8.80.
z'"i
MAGNETOSTRICTION DRIVING SYSTEM 147
due to resistance, that is, air load and support resistance. This load is
designated as the mechanical rectilineal resistance tm in Fig. 8.7CThe vibrating systems A and B given in Fig. 8.7 are usually employed
to produce sound waves in liquids or gases. The vibrating system of
Fig. 8.7C is usually employed as an element in a filter or as a frequency
standard. For the latter use it is important that the load be very small.
Or
2000 3000 4000 5000 6000FLUX DENSITY IN GAUSSES
Fig. 8.8. Magnetostriction in nickel.
The mechanical rectilineal impedance Zm at /if can be obtained from
the mechanical networks of Fig. 8.7. The motional electrical impedance
Zem can be obtained from equation 8.84. The normal electrical im-
pedance can then be determined from the electrical circuits of Fig. 8.7.
The magnetostrictive constant may be determined from the deforma-
tion-flux density characteristic. The elongation per unit length as a
function of the flux density for nickel is shown in Fig. 8.8.
The deformation per unit length, due to a force, is
X = 4^, 8.88EAwhere Jm = total force, in dynes,
A = area, in square centimeters, and
E = Young's modulus.
148 DRIVING SYSTEMS
The magnetostrictive force is
fu = KAB 8.89
where K = magnetostriction constant,
B = flux density, and
A = area, in square centimeters.
From equations 8.88 and 8.89 the deformation per unit length is
KBX = -— 8.90
The magnetostrictive constant K can be determined from the above
equation, the data of Fig. 8.8 and Young's modulus.
8.6. Piezoelectric Driving System
A piezoelectric driving system is a driving system in which the me-
chanical forces result from the deformation of a crystal having converse
1 Ze, c„^EN ..
1
PIEZOELECTRIC DRIVING SYSTEM 149
The charge, in statcoulombs, due to the application of a force, is
q = KJu 8.91
where K = constant of the crystal, 6.4 X 10~* for quartz, and
Jm = force, in dynes.
The displacement, in centimeters, due to an applied force, is
.V = — 8 92^ EA^-^
where Jm = force, in dynes,
4 = length of the crystal, in centimeters,
E = Young's modulus, and
A = cross-sectional area, in square centimeters.
From equations 8.91 and 8.92
8.93X =
150 DRIVING SYSTEMS
In the above consideration it has been assumed that the stress and
driving force are uniform over the length 4 of the crystal. Under these
conditions the crystal is a compliance given by
Cmi = 1^ 8.98EJ
wher^ ^ = cross-sectional area of the crystal, in square centimeters,
/ = length of the crystal, in centimeters, and
E = Young's modulus.
The mechanical rectilineal impedance of the crystal is
ZMi = -^7^ 8.99JuLmi
For the conditions under consideration the mechanical rectilineal
impedance of the vibrating system, from the mechanical circuit of Fig.
8.9, is
zm = zmi + zm2 8.100
where zm = total mechanical rectilineal impedance, in mechanical ohms,
zmi = mechanical rectilineal impedance of the crystal, in mechan-
ical ohms, and
zm2 = mechanical rectilineal impedance of the load, in mechanical
ohms.
The mechanical rectilineal impedances z\j, zmi and z,vf2 are referred
to one end of the crystal with the other end rigidly fixed. The dimen-
sions of the crystal are assumed to be small compared to the wavelength.
The mechanical rectilineal impedance at the end of the crystal is
fuZM = — 8.101
X
From equations 8.97, 8.100 and 8.101
/,2
ZEU = Z^2e-2^2 ^^ 8.102
where Zem = motional electrical impedance, in statohms, and
Zm — total mechanical rectilineal impedance including the crys-
tal.
PIEZOELECTRIC DRIVING SYSTEM 151
The normal electrical impedance of the crystal system is
Zen =^ ,
„ 8.1031 + JCoLeiZem
where zem = motional impedance, equation 8.102, and
Cei = capacitance of the crystal in the absence of motion.
The damped impedance and the motional impedance are effectively
in parallel as shown by equation 8.103 and depicted by the electrical
circuit in Fig. 8.9.
In the above considerations the length of the crystal is assumed to be
a small fraction of the wavelength. In general, piezoelectric driving
systems are operated at resonance. The three most common systems '
are as follows: a crystal fixed on one end and loaded on the other, a
crystal free on one end and loaded on the other and a free crystal. Thelumped constant representations of the three systems depicted by the
mechanical networks in Fig. 8.10 are valid near the resonant frequency
of the crystal.
The mass mi, in Fig. 8.10, is given by
»2i =^ 8.104
where p = density of the crystal, in grams per cubic centimeter,
4 = length of the crystal, in centimeters, and
^ = cross-sectional area of the crystal, in square centimeters.
The compliance Cmi, in Fig. 8.10, is given by
84Cmi = -T^z 8.105
where // = cross-sectional area of the crystal, in square centimeters,
le = length of the crystal, in centimeters, and
E = Young's modulus.
The compliance given by equation 8.105 is S/tt^ times the static com-pliance given by equation 8.98.
^ Mason, "Electromechanical Transducers and Wave Filters," D. Van Nos-trand Co., New York, 1942.
152 DRIVING SYSTEMS
The load on the end of the crystal is the mechanical rectilineal im-
pedance Zm2- In the case of a free crystal, Fig. 8. IOC, the only load is
the dissipation due to resistance, that is, air load and support resistance.
This load is designated as the mechanical rectilineal resistance Tm in
Fig. 8. IOC.
Cei
_ Cmi ^yi z
f "V
'Ml._ifc;
z
—
7^^4c
Ze.
El l*-EI
VIBRATING SYSTEMS MECHANICAL NETWORKS ELECTRICAL NETWORKS
Fig. 8.10. Piezoelectric driving systems. A. Crystal fixed on one end and loaded on the
other. B. Crystal free on one end and loaded on the other. C. Free crystal, that is,
a light load on both ends. In the electrical networks: zen, the normal electrical imped-
ance of the crystal, zem, the motional electrical impedance of the crystal. z_bi, the
damped electrical impedance of the crystal, zei = \/Jo>Cei- Cei, the damped electrical
capacitance of the crystal. In the mechanical networks;/.if , the driving force. zm2, the
mechanical rectilineal impedance of the mechanical load, zmi, the mechanical rectihneal
impedance of the crystal, mi and Cui, the effective mass and comphance of the crystal.
The vibrating systems ^ and B in Fig. 8.10 are usually employed to
produce sound waves in liquids or gases. The vibrating system of Fig.
8. IOC is usually employed as an element in a filter or as a frequency
standard. For the latter use it is important that the load be very small.
The mechanical rectilineal impedance zm at/,if can be obtained from
the mechanical networks of Fig. 8.10. The motional electrical impedance
Zem can be obtained from equation 8.102. The normal electrical im-
pedance can then be determined from the electrical networks of Fig.
8.10.
CHAPTER IX
GENERATING SYSTEMS
9.1. Introduction
A mechanical electrical generating system is a system for converting
mechanical or acoustical vibrations into the corresponding electrical
variations. The most common generating systems in use to-day for con-
verting mechanical vibrations into the corresponding electrical varia-
tions are the electrodynamic, the electromagnetic, the electrostatic, the
piezoelectric and the magnetostriction. It is the purpose of this chapter
to describe the electrical and mechanical characteristics of these generat-
ing systems.
9.2. Electrodynamic Generating System
A moving conductor or a moving coil generating system is a gener-
ating system in which the electromotive force is developed by motion
of a conductor through a magnetic field.
The voltage, in abvolts, due to the motion of the conductor in the
magnetic field. Fig. 9.1, is
e = Blx 9.1
where B = flux density, in gausses,
/ = length of the conductor, in centimeters, and
X = velocity of the conductor, in centimeters per second.
The velocity of the conductor is governed by the mechanical driving
force, the mechanical rectilineal impedance of the mechanical system,
and the mechanical rectilineal impedance due to the electrical system.
The vibrating system is shown in Fig. 9.1. In the mechanical circuit ^
zm represents the mechanical rectilineal impedance of the mechanical
portion of the vibrating system actuated by/if including the mechanical
rectilineal impedance of the coil at the voice coil. Jm represents the
^See footnote 1, page 125.
153
154 GENERATING SYSTEMS
mechanomotive force at the voice coil. The mechanical rectilineal im-
pedance due to the electrical system from equation 8.7 of the chapter on
Driving Systems, is
{Blfzme =
Ze9.2
where B = flux density, in gausses,
/ = length of the conductor, in centimeters,
Ze = Z£l + Ze2,
Zei = electrical impedance of the voice coil, in abohms, and
ze2 = electrical impedance of the external load, in abohms.
MAGNET
MECHANICAL CIRCUIT
(-WV 'WS^
CROSS-SECTIONAL VIEW ELECTRICAL CIRCUIT
Fig. 9.1. Electrodynamic generating system. In the mechanical circuit:/^/, the external
driving force, zi/, the total mechanical rectilineal impedance of the mechanical portion
of the vibrating system actuated by/,if . zme, the mechanical rectilineal impedance dueto the electrical circuit. In the electrical circuit: e, the internal electromotive force
generated in the voice coil, z^i, the damped electrical impedance of the voice coil.
ZA'i = TEi + joiLi. Li and tei, the damped inductance and electrical resistance of the
voice coil. Zei, the electrical impedance of the external load.
The velocity of the voice coil is
Jmzm + Zme
9.3
From equations 9.1 and 9.3 the generated electromotive force, in ab-
volts, is
BI/m= B/x =
2a/ + Zme9.4
ELECTROMAGNETIC GENERATING SYSTEMS 155
The generated electromotive force is effectively in series with the elec-
trical impedance zei of the voice coil and the electrical impedance 2^2
of the external load, as depicted by the electrical circuit in Fig. 9.1.
9.3. Electromagnetic Generating Systems
A magnetic generating system is a generating system in which the
electromotive force is developed by the charge in magnetic flux through
a stationary coil. There are two general types of magnetic generating
systems; namely, the reed armature type and the balanced armature type.
ARMATURE
MAGNET
"1COIL
n -iS
156 GENERATING SYSTEMS
where N = number of turns in the coil,
M = magnetomotive force, in gilberts, due to the steady field,
^ = area of the pole, in square centimeters,
a = spacing between the armature and pole, in centimeters, and
X = velocity of the armature, in centimeters per second.
The velocity of the armature is governed by the mechanical driving
force, the mechanical rectilineal impedance of the mechanical system,
and the mechanical rectilineal impedance due to the electrical system.
The vibrating system is shown in Fig. 9.2. In the mechanical circuit
zm represents the mechanical rectilineal impedance of the mechanical
portion of the vibrating system actuated by/:vf including the mechanical
rectilineal impedance of the armature, fu represents the mechanomo-
tive force at the armature. The mechanical rectilineal impedance due to
the electrical system from equation 8.38 of the chapter on Driving Sys-
tems is
zme = "i
—
9.6a ze
where <^i = total flux, in maxwells, through the armature,
N = number of turns on the coil,
a = spacing between the armature and pole, in centimeters, and
2£ = Z£l + ZE2,
zei = electrical impedance of the coil, in abohms, and
Ze2 = electrical impedance of the external load, in abohms.
The velocity of the armature, in centimeters per second, is
9.8
The generated electromotive force is effectively in series with the elec-
trical impedance Zei of the coil and the electrical impedance ze2 of the
external load as depicted by the electrical circuit in Fig. 9.2.
ELECTROMAGNETIC GENERATING SYSTEMS 157
B. Balanced Armature Generating System.—In the simple generating
system of the preceding section both the steady magnetic flux and the
change in flux, due to the deflection of the armature, flows through the
armature. Consider a balanced armature type of generating system in
which only the alternating flux flows longitudinally through the arma-
ture as shown in Fig. 9.3.
r~
158 GENERATING SYSTEMS
The vibrating system is shown in Fig. 9.3. In the mechanical circuit
Zm represents the mechanical rectilineal impedance of the mechanical
portion of the vibrating system including the mechanical rectilineal im-
pedance of the armature. Jm represents the mechanomotive force on the
armature. The mechanical rectilineal impedance due to the electrical
system from equation 8.52 of the chapter on Driving Systems is
^ME =2r- — ^-^^AN^Ma^R2ZE
where A'^ = number of turns in the coil,
(^ = total flux in the air gap at one of the poles, in maxwells,
M. = magnetomotive force, in gilberts, of the magnet,
a = spacing between armature and pole, in centimeters,
i?2 = reluctance, in oersteds, of the alternating magnetic circuit,
ZE = Zei + Ze2,
zei = electrical impedance of the coil, in abohms, and
ze2 = electrical impedance of the external load, in abohms.
The velocity of the armature, in centimeters per second, is
/mZm + zme
From equations 9.9 and 9.11
9.11
NMJ NMAJme = X = "^— 9.12
a^ a^{zM + zme)
The generated electromotive force is effectively in series with the
electrical impedance Ze\ of the coil and the electrical impedance ze2 of
the external load, as depicted by the electrical circuit in Fig. 9.3.
9.4. Electrostatic Generating System
A condenser or electrostatic generating system is a generating system
in which the electromotive force is developed by the relative motion
between two differently electrostatically charged plates.
The current, in statamperes, generated by the motion of the movableplate of the condenser from equation 8.67 of the chapter on Driving
Systems is ^.
ELECTROSTATIC GENERATING SYSTEM 159
where e = polarizing voltage, in statvolts,
^ = area of the plate, in square centimeters,
a = spacing between the plates, in centimeters, and
X = velocity of the movable plate, in centimeters per second.
The current, in statamperes, due to the electromotive force e across
the electrical impedances z^i and z_b2 of Fig. 9.4 is
t2
e9.14
where e = electromotive force, in statvolts,
Z£iZi'2Ze
1
2J51 =JuiCei
Ce\ = electrical capacitance of the condenser, in statfarads, and
Ze2 = electrical impedance of the external load, in statohms.
r ^p-W»-l|l|l|l|i
DIAPHRAGM
CErTL
.stationary"plate
J
1
160 GENERATING SYSTEMS
and 9.14 the generated electromotive force, e, in statvolts, across the
electrical impedances zei and Ze2 is
CqAxe = -—^ zjj 9.15
The velocity of the movable plate is governed by the mechanical
driving force, the mechanical rectilineal impedance of the mechanical
system and the mechanical rectilineal impedance due to the electrical
system. The vibrating system is shown in Fig. 9.4. In the mechanical
circuit Zm represents the mechanical rectilineal impedance of the mechan-
ical portion of the vibrating system actuated by /m including the
mechanical impedance of the movable plate, /m represents the mechano-
motive force at the movable plate. The mechanical rectilineal impe-
dance due to the electrical system from equation 8.71 of the chapter on
Driving Systems is
16rZME = ,^2 4 ^E 9.16
where eo = polarizing voltage, in statvolts,
a = spacing between plates, in centimeters,
A = area of the plates, in square centimeters.
Ze =Z£l + 'ZE2
1
Ce\ = capacity of the generator, in statfarads,
Ze2 = electrical impedance of the external load, in statohms.
The velocity of the movable plate, in centimeters per second, is
/^^X =
Zm + zme9.17
From equations 9.15 and 9.17 the electromotive force e across z^i and
Ze2 in parallel, depicted by the electrical network of Fig. 9.4, is
^9.18
\ira^{%M + Zme)
ELECTROSTATIC GENERATING SYSTEM 161
The electromotive force ei in series with z^i and Z£;2 which will pro-
duce the electromotive force e across ze2 is of interest in the design of
generating systems.
Equation 9.15 may be written
^^epJx/ ZE,ZE2 \g^g
4irfl^ \Zei + Ze2/
The electrical capacitance of the condenser Cei from equation 8.62
of the chapter on Driving Systems is
ACei = -— 9.20
The electrical impedance ze\ is
1 47r«7.E1 = -r—- = ^^ 9.21
jwLei JoiA
Substituting equation 9.21 in 9.19,
epx ( Ze2 \ajw \zei + Ze2/
The amplitude in terms of the velocity is
9.22
XX = — 9.23
Substituting equation 9.23 in 9.22,
\Ze1 + 2£2/. = ^(---^) 9.24
a
The electromotive force e in terms of e\ and the impedances Ze\ and
ZE2 is
e\ZE2
Comparing equations 9.24 and 9.25,
9.25
CnXfi = — 9.26
a
162 GENERATING SYSTEMS
The electrostatic generating system may be considered to consist of
a generator having an internal or open circuit electromotive force ei as
given by equation 9.26 and an internal impedance zei. Equation 9.26
shows that this electromotive force is independent of the frequency if
the amplitude is independent of the frequency. However, the voltage e
across the load may vary with frequency depending upon the nature of
load Z£2-
9.5. Magnetostriction Generating System
A magnetostriction generating system is a generating system in which
the electromotive force is developed in a stationary coil by a change in
fM
MECHANICAL CIRCUIT
MAGNET
/{JT"
'ei '-I
r^M 'SW—
j
ELECTRICAL CIRCUIT
Fig. 9.5. Magnetostriction generating system. In the mechanical circuit: zm, the total
mechanical rectilineal impedance of the mechanical portion of the vibrating systemactuated by/iif. zme, the mechanical rectilineal impedance due to the electrical circuit.
In the electrical circuit: e, the internal electromotive force generated in the coil, zei, the
damped electrical impedance of the coil, z^i = r_Bi + juLi. L\ and rE\, the dampedinductance and electrical resistance of the coil. ZEi, the electrical impedance of the
external load.
magnetic flux due to the deformation of a ferromagnetic material having
magnetostriction properties. The magnetostriction generator, shown in
Fig. 9.5, consists of a coil surrounding a magnetic circuit which includes
a ferromagnetic material having magnetostriction properties. The volt-
age, in abvolts, developed in the coil due to deformation of the rod,
from equation 8.78 of the chapter on Driving Systems, is
^tNK9.27
MAGNETOSTRICTION GENERATING SYSTEM 163
where N = number of turns in the coil,
R = reluctance of the magnetic circuit,
K = constant representing the dynamical Villari magnetostric-
tion effect, and
X = velocity at the point of application of the driving force to
the rod, in centimeters per second.
The velocity of the rod is governed by the mechanical driving force,
the mechanical impedance of the mechanical system and the mechanical
impedance due to the electrical system. The vibrating system is shown
in Fig. 9.5. In the mechanical circuit zm represents the mechanical
rectilineal impedance of the mechanical portion of the vibrating system
actuated by /m including the mechanical rectilineal impedance of the
magnetostriction rod. /m represents the mechanomotive force on the
rod. It is assumed that the force/m is the same at all points along the
length of the rod and that the phase of the amplitude is constant along
the rod. The mechanical rectilineal impedance due to the electrical sys-
tem from equation 8.79 or 8.84 of the chapter on Driving Systems is
where A'^ = number of turns in the coil,
K = magnetostriction constant,
R = reluctance of the magnetic circuit, and
Ze = Z£l + Z£2)
Zei = electrical impedance of the coil, in abohms, and
Ze2 = electrical impedance of the external circuit, in abohms.
The dimensions of the rod are assumed to be a small fraction of a
wavelength. Under these conditions the rod is a compliance given by
Cu. - £ 9.29
where A = cross-sectional area of the rod, in square centimeters,
/ = length of the rod, in centimeters, and
E = Young's modulus.
164 GENERATING SYSTEMS
The mechanical rectilineal impedance of the rod is
1
zmi = T-—
—
9.30jwCmi
For the conditions under consideration the mechanical rectilineal im-
pedance of the vibrating system is
zm = zmi + zm2 9.31
where Zm = total mechanical rectilineal impedance, in mechanical ohms,
Zmi = mechanical rectilineal impedance of the rod, in mechanical
ohms, and
zji/2 = mechanical rectilineal impedance of the load, in mechanical
ohms.
The velocity of the rod, in centimeters per second, at the driving
point is
X = -^^^- 9.32Zm + zme
From equations 9.27 and 9.32 the generated electromotive force, in
abvolts, is
iirNK/M
(zm + zme)R9.33
The generated electromotive force is effectively in series with the
electrical impedance Zgi of the coil and the electrical impedance ze2 of
the external load as depicted by the electrical circuit in Fig. 9.5.
In the above considerations the length of the rod is assumed to be a
small fraction of the wavelength. In general, magnetostriction gener-
ating systems are operated at resonance. The two most common sys-
tems are as follows: a rod fixed on one end and driven on the other and
a rod free on one end and driven on the other. The lumped constant
representations of the two systems shown in Fig. 9.6, are valid near the
resonant frequency of the rod. The mass m-i and compliance Cm\, in
Fig. 9.6, are given by equations 8.86 and 8.87 in the chapter on Driving
Systems. The load on the end of the rod is the mechanical rectilineal
impedance zm2- The mechanical rectilineal impedance zme due to the
electrical circuit is given by equation 9.28. From the above constants
and the driving force /u the velocity in the mechanical circuit can be
PIEZOELECTRIC GENERATING SYSTEM 165
determined. The open circuit electromotive force e of the electrical
circuit of Fig. 9.6 can be obtained from equation 9.27 and the velocity.
z 1"
ZMl*f
^El*<^
T
Zmi-*-i
Emi m.
'El**^-l
e
I
VIBRATING SYSTEMS MECHANICAL NETWORKS ELECTRICAL CIRCUITS
Fig. 9.6. Magnetostriction generating systems. A. Rod fixed on one end and driven on
the other. B. Rod free on one end and driven on the other. In the mechanical net-
works: /m, the driving force, zme, the mechanical rectilineal impedance due to the
electrical circuit. ZMi, the mechanical rectilineal impedance of the external mechanical
load. Zjv/i, the mechanical rectilineal impedance of the rod. mi and Cmi, the effective
mass and compliance of the rod. In the electrical circuits: e, the internal electromotive
force generated in the coil, zei, the damped electrical impedance of the coil. 2_bi =^Ei + j<^Li. Li and tei, the damped inductance and electrical resistance of the coil, zei,
the electrical impedance of the external electrical load.
9.6. Piezoelectric Generating System
A piezoelectric generating system is a generating system in which the
electromotive force is developed by the deformation of a crystal having
converse piezoelectric properties. The crystal generating system, shownin Fig. 9.7, consists of a suitably ground crystal having converse piezo-
electric properties fitted with appropriate electrodes.
The current, in statamperes, generated by the motion of the crystal
from equation 8.94 of the chapter on Driving Systems is
KEJh = —.— •* 9.34
where K = constant of the crystal, 6.4 X 10~* for quartz,
E = Young's modulus,
4 = length of the crystal, in centimeters,
^ = cross-sectional area of the crystal, in square centimeters,
length of the crystal, in centimeters, and
X = velocity of the crystal, in centimeters per second.
166 GENERATING SYSTEMS
The current, in statamperes, due to the electromotive force e across
the electrical impedances z^i and Ze2 is
i2e
Ze9.35
where e = electromotive force, in statvolts.
2£
ZEI
Zei + ZE2
I
j'wCei
Cei = electrical capacitance of the crystal, in statfarads, and
2e2 = electrical impedance of the external load, in statohms.
ELECTRODE
[ffiCji /^7777777777777
MECHANICAL CIRCUIT
re X
ELECTRICAL NETWORK
Fig. 9.7. Piezoelectric generating system. In the electrical circuit: zu, the total mechan-ical rectilineal impedance of the mechanical portion of the vibrating system actuated by
}m- 2me, the mechanical rectilineal impedance due to the electrical network. In the
electrical circuit: e, the electromotive force generated across the crystal, zei, the
damped electrical impedance of the crystal, z^'i = \/Jo}Ce\- Ce\, the damped elec-
trical capacitance of the crystal. 2_b2, the electrical impedance of the external load.
Since there is no external current applied to the electrodes of the
crystal the sum of the currents ii and ii is zero. From equations 9.34
and 9.35 the generated electromotive force e, in statvolts, across the
electrical impedance Zei and Ze2 is
KEAxe = —-— ZE 9.36
The velocity at the end of the crystal is governed by the mechanical
driving force, the mechanical rectilineal impedance of the mechanical
PIEZOELECTRIC GENERATING SYSTEM 167
system and the mechanical rectilineal impedance due to the electrical
system. The vibrating system is shown in Fig. 9.7. In the mechanical
circuit Zm represents the mechanical rectilineal impedance of the mechan-
ical portion of the vibrating system actuated by Jm including the
mechanical rectilineal impedance of the crystal, /m represents the
mechanomotive force at the end of the crystal. It is assumed that the
force/iif is the same at all points along the length of the crystal and that
the phase of the amplitude is constant along the crystal. The mechani-
cal impedance due to the electrical system from equation 8.97 or 8.102
of the chapter on Driving Systems is
zme = 7^— ze 9.37
where K = constant of the crystal 6.4 X 10~* for quartz,
E = Young's modulus,
4 = length of the crystal, in centimeters,
A = area of the electrode, in centimeters, length of the crystal,
in centimeters, and
ZeiZe2 „ „„Ze = 9.38
zei + zj;2
ZEi = —TT- 9.39Jo:Lei
Cei = capacitance of the generator, in statfarads,
Ze2 = electrical impedance of the external load, in statohms.
The dimensions of the crystal are assumed to be a small fraction of a
wavelength. Under these conditions the crystal is a compliance given
by
Cmi = ^ 9.40EA
where A = cross-sectional area of the crystal, in square centimeters,
4 = length of the crystal, in centimeters, and
E = Young's modulus.
The mechanical rectilineal impedance of the crystal is
ZMi = ^-^r- 9.41JwLmx
168 GENERATING SYSTEMS
For the conditions under consideration the mechanical rectilineal im-
pedance of the vibrating system is
zm = ZiWi + zm2 9.42
where Zm = total mechanical rectilineal impedance, in mechanical ohms,
zmi = mechanical rectilineal impedance of the crystal, in mechan-
ical ohms, and
zm2 = mechanical rectilineal impedance of the load, in mechanical
ohms.
The velocity at the end of crystal, in centimeters per second, is
X = --^^ 9.43Zm + zme
From equations 9.36 and 9.43 the electromotive force across z^i and
Ze2 in parallel, depicted by the electrical network of Fig. 9.7, is
. = ^^^'^f-^9.44
4(zm + Zme)
The electromotive force ei in series with Zei and Ze2 which will pro-
duce the electromotive force e across Ze2 is of interest in the design of
generating systems.
Equation 9.36 may be written
KEJx / zeiZe2 \ „ . fe = —— ( ~-
1 9.454 \zei + Ze2/
The electrical capacitance of the crystal is
JD^- = i;4
'''
where D = dielectric constant of the crystal.
The electrical impedance z^i is
1 4ir4"El =
j<j>Cei jo^AD9.47
PIEZOELECTRIC GENERATING SYSTEM 169
Substituting 9.47 in 9.45,
e =AirKEx / ze2 \
jwB \ze\ + Ze2/9.48
The amplitude in terms of the velocity is
XX = ~ 9.49
Substituting 9.49 in 9.48,
e
AttKEx / Ze2 \
D \zei + Ze2/9.50
The electromotive force e in terms of e^ is
, = _^i^_ 9.51ZEI + Ze2
Comparing equations 9.50 and 9.51,
AwKExe. == -^ 9.52
The piezoelectric generating system may be considered to consist of
a generator having an internal or open circuit electromotive force e\ as
given by equation 9.52 and an internal impedance Ze\. Equation 9.52
shows that this electromotive force is independent of the frequency if
the amplitude is independent of the frequency. However, the voltage
e across the load may vary with frequency depending upon the nature
of the load Ze2-
In the above considerations the length of the crystal is assumed to be
a small fraction of the wavelength. In general piezoelectric generating
systems are operated at resonance. The two most common systems are
as follows: a crystal fixed on one end and driven on the other and a
crystal free on one end and driven on the other. The lumped constant
representations of the two systems shown in Fig. 9.8 are valid near the
resonant frequency of the crystal. The mass m\ and compliance Cmuin Fig. 9.8, are given by equations 8.104 and 8.105 in the chapter on
Driving Systems. The load on the end of the crystal is the mechanical
rectilineal impedance 2:^/2- The mechanical rectilineal impedance zme
170 GENERATING SYSTEMS
due to the electrical circuit is given by equation 9.37. From the above
constants and the driving force /,vf the velocity in the mechanical cir-
cuit can be determined. The electromotive e across the crystal of the
*-MI
Zme
J XCjIze, r^
i_XVIBRATING SYSTEMS MECHANICAL NETWORKS ELECTRICAL NETWORKS
Fig. 9.8. Piezoelectric generating systems. A. Crystal fixed on one end and driven on
the other. B. Crystal free on one end and driven on the other. In the mechanical net-
works: Jm, the driving force, zme, the mechanical rectilineal impedance due to the
electrical circuit. Zm2, the mechanical rectilineal impedance of the external mechanical
load. Ziifi, the mechanical rectilineal impedance of the crystal. OTi and Cmi> the
effective mass and compliance of the crystal. In the electrical circuits: e, the electro-
motive force generated across the crystal. ze\, the damped electrical impedance of the
crystal, zei = \/joiCei- Cei, the damped electrical capacitance of the crystal, zei,
the electrical impedance of the external load.
electrical network of Fig. 9.8 can be obtained from equation 9.45 and the
velocity.
CHAPTER X
THEOREMS
10.1. Introduction
A number of dynamical laws common to electrical, mechanical recti-
lineal, mechanical rotational and acoustical systems have been described
in this book. There are other dynamical laws that are well known in
electrical circuit theory which can be applied to mechanical and acousti-
cal systems. It is the purpose of this chapter to illustrate the applica-
tion of reciprocity, Thevenin's and superposition theorems to electrical,
mechanical rectilineal, mechanical rotational and acoustical systems.
10.2. Reciprocity Theorems '
A. Electrical Reciprocity Theorem.—In an electrical system composed
of the electrical elements of inductance, electrical capacitance and elec-
trical resistance, let a set of electromotive forces ^i , ^2 > ^3 • ^n all
harmonic of the same frequency acting in n points in the invariable net-
work, produce a current distribution z'l', /2') iz in , and let a second
set of electromotive forces ex" , e^' , e^" . . . e^' of the same frequency as
the first produce a second current distribution zi", i^' , /s" . . . z„". Then
n n
10.1
This theorem is valid provided the electrical system is invariable,
contains no internal source of energy or unilateral device, linearity in
the relations between electromotive forces and currents and complete
reversibility in the elements, and provided the electromotive forces
^1) ^2) ^3 • • . ^n are all of the same frequency.
1 Ballentine, S., Proc. I.R.E., Vol. 17, No. 6, p. 929, 1929. "Reciprocity in
Electromagnetic and Other Systems."
171
172 THEOREMS
In the simple case in which there are only two electromotive forces,
as illustrated in the electrical system of Fig. 10.1, equation 10.1 becomes
e'i" = /'/' 10.2
where e' , e" and /', /" are the electromotive forces and currents depicted
in the electrical system of Fig. 10.1.
oe
RECIPROCITY THEOREMS 173
the elements, and provided the applied forces /mi, /ji/2,/m3 • • -JMn are
all of the same frequency.
In the simple case in which there are only two forces, as illustrated in
the mechanical rectilineal system of Fig. 10.1, equation 10.3 becomes
fu'x" =/a/V 10.4
where /^f', /m" and x' , x" are the forces and velocities depicted in the
mechanical rectilineal system of Fig. 10.1.
C. Mechanical Rotational Reciprocity Theorem.—In a mechanical rota-
tional system composed of mechanical rotational elements of momentof inertia, rotational compliance and mechanical rotational resistance,
let a set of torques/iji',/7j2', //ea' . • Jun all harmonic of the same fre-
quency acting in n points in the system, produce a rotational velocity
distribution ^i', ^2', 4>2,' • • '<i>n , and let a second set of torques Jrx' ,
fB2",fRi" • Jru" of the same frequency as the first produce a second
rotational velocity distribution <j!>i", 4>2" , '4>^". . . <l>n" Then
n n
j= 1 y= 1
This theorem is valid provided the mechanical rotational system is
invariable, contains no internal source of energy or unilateral device,
linearity in the relations between torques and rotational velocities, andprovided the applied torques /iji,/ij2,/fl3 . . ./ij„ are all of the samefrequency.
In the simple case in which there are only two torques, as illustrated
in the mechanical rotational system of Fig. 10.1, equation 10.5 becomes
fR'^"=fR'4>' 10.6
where /ij ,/b' and (J)', fli" are the torques and angular velocities depicted
in the mechanical rotational system of Fig. 10.1.
D. Acoustical Reciprocity Theorem?—From equation ^ 1.4 of "TheElements of Acoustical Engineering"
dv 1— + -grad^o = 10.7at p
^ Rayleigh, "Theory of Sound," Vol. II, p. 145, MacMillan and Co., London,1926.
^ Olson, "Elements of Acoustical Engineering," D. Van Nostrand Co., NewYork, 1940.
174 THEOREMS
Consider two independent sets of pressures p',p" and particle veloci-
ties v' and v" . Multiply equation 10.4 by the p and v of the other set.
,, dv' , dv" 1 ,, , , 1 , , „y" — - 0' —- + - v" grad po - - v' grad po = 10.8
dt dt p p
If p and V vary as a harmonic of the time, equation 10.5 becomes
v' grad po" - -v' grad po" = 10.9P
There is the following relation:
V grad^ = div vp — p div v 10.10
From equations 1.9 and 1.10 of "The Elements of Acoustical Engi-
neering"
^ + divy = 10.11ypo ot
From equations 10.8, 10.9 and 10.10
div {v"pq' - v'pq") =0 10.12
The relation of equation 10.12 is for a point. Integration of equation
10.12 over a region of space gives
// {v"pq' - v'pQ")ds = 10.13
If, in an acoustical system comprising a medium of uniform density
and propagating irrotational vibrations of small amplitude, a pressure
p' produces a particle velocity v' and a pressure p" produces a particle
velocity v", then
iv"p' - v'p")nds = 10.14//where the surface integral is taken over the boundaries of the volume.
In the simple case in which there are only two pressures, as illustrated
in the free field acoustical system of Fig. 10.1, equation 10.14 becomes
^V' = p"v' 10.15
where p',p" and v' , v" are the pressures and particle velocities depicted
in the free field acoustical system of Fig. 10.1.
RECIPROCITY THEOREMS 175
The above theorem is applicable to all acoustical problems. However,
the above theorem can be restricted to lumped constants as follows:
In an acoustical system composed of inertance, acoustical capacitance
and acoustical resistance let a set of pressures p\ , p2 , pz • • Pn all har-
monic of the same frequency acting in n points in the system, produce a
volume current distribution Xi, X2, X^ . . . X^, and let a second set of
pressures ^i",^2"j/'3" pn." of the same frequency as the first, produce
a second volume current distribution Xi", X2", X^" . . . Xn" Then
n n
Y^Pi'xr =Y^prxi 10.16
3= 1 i= 1
This theorem is valid provided the acoustical system is invariable,
contains no internal source of energy or unilateral device, linearity in
the relations between pressures and volume currents and complete
reversibility in the elements, and provided the applied pressures p\, pi,
pz . . . pn are all of the same frequency.
In the simple case in which there are only two pressures, as illustrated
in the acoustical system of lumped constants in Fig. 10.1, equation 10.16
becomes
p'X" = p"X' 10.17
where p',p" and X' , X" are the pressures and volume currents depicted
in the acoustical system of lumped constants in Fig. 10.1.
E. Mechanical-Acoustical Reciprocity Theorem.—In an interconnected
mechanical-acoustical system let a set of forces/if i' . . . JMn act in the
mechanical system, and a set of pressures pi' . . . pn act in the acousti-
cal system with the resultant velocities Xx . . . xj in the mechanical
system and with the resultant volume currents Xi . . . Xn in the
acoustical system; let also,/", x",p" and X" represent a second set of
such forces, velocities, pressures and volume currents. Then
n n
2] (A^/*/' + Pi^i') = Z) (/m/'^Z + P/'X/) 10.18
J=
1
3= 1
In the simple case in which there is only one force in the mechanical
system and one pressure in the acoustical system
/m'x" = p"X' 10.19
176 THEOREMS
Equation 10.19 states that if a unit force/jif' in the mechanical system
produces a certain volume current X' in the acoustical system, then a
unit pressure p" acting in the acoustical system will produce a velocity
x" in the mechanical system which is numerically the same as the volume
current previously produced in the acoustical system.
The mechanical-acoustical reciprocity theorem is illustrated in Fig.
10.2^.
P"X'
B
L.S.
tn\P"X'
i" L.S. M ., px L.S.
HX*p"X
D EFig. 10.2. Reciprocity in the following systems depicted above:
A. Mechanical-acoustical.
B. Electrical-mechanical.
C. Electrical-mechanical-acoustical.
D. Electrical-mechanical-acoustical-mechanical-electrical.
E. Acoustical-mechanical-electrical-mechanical-acoustical.
F. Electrical-Mechanical Reciprocity Theorem.—In an interconnected
electrical-mechanical system let a set of electromotive forces e\ . . . en
act in the electrical system, and a set of forces /mi' . . fun act in the
mechanical system with the resultant currents i\ . . . in in the electrical
system and with the resultant velocities X\ . . . Xn in the mechanical
system; let also, e" , i",Jm" and x" represent a second set of electro-
motive forces, currents, forces and velocities. Thenn n
Y^ iejil' + Jm/x/') = 23 {e/'i/ + fu/'x/) 10.20
y= 1 j= 1
In the simple case in which there is only one electromotive force in the
electrical system and one force in the mechanical system
e'i" = Jm"x' 10.21
RECIPROCITY THEOREMS 177
Equation 10.21 states that if a unit electromotive force e in the electri-
cal system produces a certain velocity x' in the mechanical system, then
a unit force /iif" in the mechanical system will produce a current i" in
the electrical system which is numerically the same as the velocity pre-
viously produced in the mechanical system.
The electrical-mechanical reciprocity theorem is illustrated in Fig.
10.2fi.
G. Electrical-Mechanical-Acoustical Reciprocity Theorem.—Since rec-
iprocity relations hold in electrical-mechanical and mechanical-acoustical
systems, they will also hold the three systems interconnected in the
order electrical, mechanical, acoustical. This type of system embraces
practically all electroacoustic transducers.
For the simple case of a pressure p' in a sound field producing a cur-
rent i' in the electrical system and a voltage e" in the electrical system
producing a volume current X' in the sound field the reciprocity rela-
tion may be written
{p'X")ds = e"i' 10.22//'Equation 10.23 states that if, in the electrical system of a loud speaker,
a generator of electromotive force e" produces, at a point in a sound field,
a volume current X" , than a numerically equal pressure in the sound
field at that point will produce a current i' in the electrical system equal
to the previously produced volume current X" in the sound field.
The electrical-mechanical-acoustical reciprocity theorem is illustrated
in Fig. 10.2C.
H. Electrical-Mechanical-Acoustical-Mechanical-Electrical Reciprocity
Theorem.—In most cases in the reproduction of sound the original sound
is converted into electrical energy by a microphone which is usually an
acoustical, mechanical, electrical transducer. Then it is converted back
into acoustical energy by means of a loud speaker or an electrical, me-
chanical, acoustical transducer.
If both microphone and loud speaker are reversible and the air is the
connecting medium, as shown in Fig. 10.2D, and if an electromotive force
e' in the loud speaker will produce a current i' in the microphone, then
an equal electromotive force /' in the microphone will produce the samecurrent /" in the loud speaker. This may be expressed as
e'i" = e"i' 10.23
178 THEOREMS
I. Acoustical-Mechanlcal-EIectrical-Mechanical-Acoustical Reciprocity
Theorem.—If both microphone and loud speaker are reversible and the
two connected electrically, as shown in Fig. 10.2£, and if a pressure p' at
a point in the vicinity of the microphone will produce a volume current
X' at a point in the vicinity of the loud speaker, then an equal pressure
p" at the same point in the vicinity of the loud speaker will produce the
same volume current X" at the same point in the vicinity of the micro-
phone. This may be expressed as
p'X" = p"X' 10.24
10.3. Thevenin's Theorems
A. Thevenin's Electrical Theorem.—If an electrical impedance Ze be
connected between any two points in an electrical circuit, the current /
through this electrical impedance is the electromotive force e between
the points prior to the connection divided by the sum of the electrical
impedance ze and the electrical impedance ze , where ze is the electrical
impedance of the circuit measured between the two points prior to con-
necting Ze-
B. Thevenin's Mechanical Rectilineal Theorem.—If a mechanical recti-
lineal impedance Zm be connected at any point in a mechanical rectilin-
eal system, the resultant velocity of this mechanical rectilineal imped-
ance is the product of the velocity and mechanical rectilineal impedance
Zm' of the system both measured at the point prior to the connection
divided by the sum of the mechanical rectilineal impedances Zm and zm'
C. Thevenin's Mechanical Rotational Theorem.—If a mechanical rota-
tional impedance zr be connected at any point in a mechanical rota-
tional system, the resultant angular velocity of this mechanical rotational
impedance is the product of the angular velocity and mechanical rota-
tional impedance zr of the system both measured at the point prior to
the connection divided by the sum of the mechanical rotational imped-
ances Zr and Zr'.
D. Thevenin's Acoustical Theorem.—If an acoustical Impedance za
be connected at any point in an acoustical system, the volume current
X in this acoustical impedance is the pressure p at the point prior to the
connection divided by the sum of the acoustical impedance Za and the
acoustical impedance za' , where Za' is the acoustical impedance at the
point prior to connecting za-
SUPERPOSITION THEOREM 179
10.4. Superposition Theorem
Consider the simultaneous action of a number of electromotive forces,
forces, torques or pressures distributed throughout an electrical, me-
chanical rectilineal, mechanical rotational or acoustical system. Thecurrent, velocity, angular velocity or volume current at any point or the
electromotive force, force, torque or pressure at a location is the sum of
the currents, velocities, angular velocities or volume currents or electro-
motive forces, forces, torques or pressures at these locations which would
exist if each source were considered separately. Each source, save the
one being considered, must be replaced by a unit of equivalent internal
electrical, mechanical rectilineal, mechanical rotational or acoustical
impedance.
CHAPTER XI
APPLICATIONS
11.1. Introduction
The fundamental principles relating to electrical, mechanical recti-
lineal, mechanical rotational and acoustical analogies have been estab-
lished in the preceding chapters. Employing these fundamental prin-
ciples the vibrations produced in mechanical and acoustical systems due
to impressed forces may be solved as follows: Draw the electrical net-
work which is analogous to the problem to be solved. Solve the electri-
cal network by conventional electrical circuit theory. Convert the
electrical answer into the original system. In this procedure any prob-
lem involving vibrating systems is reduced to the solution of an electri-
cal network. A complete treatment of the examples of the use of
analogies in the solution of problems in mechanical and acoustical sys-
tems is beyond the scope of this book. However, a few typical examples
described in this chapter will serve to illustrate the principles and
method.
11.2. Automobile Muffler
The sound output from the exhaust of an automobile engine contains
all audible frequencies in addition to frequencies below and above the
audible range. The purpose of a muffler is to reduce the audible exhaust
sound output. An ideal muffler should suppress all audible sound which
issues from the exhaust without increasing the exhaust back pressure.
The original mufflers consisted essentially of a series of chambers
which increased progressively in volume. The idea was to allow the gases
to expand and thereby reduce the noise. Actually it was a series of
acoustical capacitances. This muffler is quite effective. However, by
the application of acoustic principles an improved muffler has been
developed in which the following advantages have been obtained:
smaller size, higher attenuation in the audible frequency range and reduc-
180
AUTOMOBILE MUFFLER 181
tion of engine back pressure. A cross-sectional view of the improved
muffler is shown in Fig. 11.1. The acoustical network ^ shows that the
system is essentially a low pass filter. The main channel is of the same
diameter as the exhaust pipe. Therefore, there is no increase in acousti-
cal impedance to direct flow as compared to a plain pipe. In order not
to impair the efficiency of the engine it is important that the muffler does
not increase the acoustical impedance to subaudible frequencies. The
system of Fig. 11.1 can be designed so that the subaudible frequencies
CROSS-SECTIONAL VIEW ACOUSTICAL NETWORK
Fig. 11.1. Cross-sectional view and acoustical network of an automobile muffler.
are not attenuated and at the same time introduce high attenuation in
the audible frequency range.
The terminations at the two ends of the network are not ideal. There-
fore it is necessary to use shunt arms tuned to different frequencies in
the low portion of the audible range. Acoustic resistance is obtained
by employing slit type openings into the side chambers.
In a development of this kind the nature of sound which issues from
the exhaust is usually determined. From these data the amount of sup-
pression required in each part of the audible spectrum may be ascer-
tained. From these data and the terminating acoustical impedances the
' In the illustrations in the preceding chapters the elements in the electrical
network have been labeled r/j, L and Ce- However, in using analogies in actual
practice the conventional procedure is to label the elements in the analogouselectrical network with r.u, m and Cm for the mechanical rectilineal system, withrft, /and Cu for a mechanical rotational system with rA,M and Ca for an acousti-
cal system. This procedure will be followed in this chapter in labeling the ele-
ments of the analogous electrical circuit. It is customary to label this networkwith the caption "analogous electrical network of the mechanical system" (or
of the rotational system or of the acoustical system) or with the caption "mechan-ical network" (or rotational network or acoustical network). The latter con-vention will be used in this chapter. When there is only one path, circuit will
be used instead of network.
182 APPLICATIONS
network can be developed. In general, changes are required to compen-
sate for approximations. In this empirical work the acoustical network
serves as a guide in directing the appropriate changes.
11.3. Electric Clipper
An electric clipper is shown schematically in Fig. 11.2. The driving
system is the unpolarized type described in section 8.3. The actuating
CUTTING HEAD/1IIIIIII III!
"eiLi.
DIRECT RADIATOR LOUD SPEAKER
11.4. Direct Radiator Loud Speaker
183
The direct radiator type loud speaker shown in Fig. 11.3 is almost
universally used for radio and phonograph reproduction. The mechani-
cal circuit of this loud speaker is also shown in Fig. 11.3. The mechani-
cal rectilineal impedance at the point /^ can be determined from the
mechanical circuit. Then the motional electrical impedance can be de-
termined from equation 8.7. The current in the voice coil can be deter-
|-A/W-AMr^KR^-j
BAFFLE
ELECTRICAL CIRCUIT
"1| ,," I"*.! <^Z W
fM
CROSS-SECTIONAL VIEW MECHANICAL CIRCUIT
Fig. 11.3. Cross-sectional view, electrical circuit and mechanical circuit of a direct
radiator dynamic loud speal<er. In the electrical circuit: e, the open circuit voltage of
the generator or vacuum tube, teg, the electrical resistance of the generator or vacuumtube. rEi and Li, the electrical resistance and inductance of the voice coil, zem, the
motional electrical impedance of the driving system. In the mechanical circuit: m\,
the mass of the cone. rMi and Ci, the mechanical rectilineal resistance and compHanceof the suspension, nii and r.w2, the mass and mechanical rectilineal resistance of the air
load.
mined from the electrical circuit of Fig. 11.3. The mechanical driving
force can be determined from equation 8.1. The velocity can be deter-
mined from the mechanical circuit of Fig. 11.3 as follows:
/mZmt
11.1
where zmt = total mechanical impedance at the point/if , in mechanical
ohms, and
Jm = driving force, in dynes.
184 APPLICATIONS
The sound power output, in ergs, is given by
P = rMX^ 11.2
where tm = mechanical radiation resistance, in mechanical ohms,
X = velocity of the cone, in centimeters per second.
The object ^ is to select the constants so that the power output as
given by equation 11.2 is independent of the frequency over the desired
frequency range.
11.5. Rotational Vibration Damper
In reciprocating engines and other rotating machinery rotational
vibrations of large amplitude occur at certain speeds. These rotational
, FX.YWHEEL I,MOMENT OFINERTIA I2
COMPLIANCE Cn
C -^
FRICTION/BEARING '
RESISTANCETo
2fqit:Hfe-
END VIEW SIDE VIEW ROTATIONAL NETWORK
Fig. 11.4. Flywheel and vibration damper. In the mechanical network: A, the momentof inertia of the flywheel. li the moment inertia of the damper. Cg, the rotational
comphance of the damper, m, the mechanical rotational resistance between the damperand the shaft.
vibrations are sometimes of such high amplitude that the shafts will fail
after a few hours of operation. A number of rotational vibration damp-
ers have been developed for reducing these rotational vibrations. It
is the purpose of this section to describe one of these systems for con-
trolling the vibrations in a rotational system. The simple vibration
damper, shown in Fig. 11.4, is used to control the vibrations of the fly-
^ For a specific description of and expressions for the elements of the mechani-cal system see H. F. Olson, "Elements of Acoustical Engineering," D. VanNostrand Co., New York, 1940. In this book all types of acoustical vibrating
systems are analyzed by the use of analogies. These systems include micro-phones, loud speakers, phonograph pickups, telephone receivers, measuringsystems, etc.
MACHINE VIBRATION ISOLATOR 185
wheel. The damper consists of a moment of inertia I2 rotating on a
shaft with a mechanical rotational resistance tr. The moment of inertia
is coupled to the flywheel by a spring of compliance Cm- The rotational
compHance is Cr = CmIc? , where a is the radius at the point of attach-
ment of the spring. This system forms a shunt mechanical rotational
system. The shunt mechanical rotational circuit is tuned to the fre-
quency of the vibration. Since the mechanical rotational impedance of
a shunt resonant rotational circuit is very high at the resonant frequency
the angular velocity of the vibration of the flywheel will be reduced. Aconsideration of the mechanical rotational network illustrates the prin-
ciple of the device. This is one example of the many types of vibration
dampers for use in absorbing rotational vibrations. The action of these
systems may also be analyzed by the use of analogies.
11.6. Machine Vibration Isolator
The vibration of a machine is transmitted from its supports to all
parts of the surrounding building structure. In many instances this
MECHANICALRECTILINEAL
SYSTEMMECHANICAL
CIRCUITMECHANICALNETWORK
Fig. 11.5. Machine vibration isolator. A. Machine mounted directly upon the floor.
In the mechanical rectilineal system and mechanical circuit: mi, the mass of the machine.zmf, the mechanical rectilineal impedance of the floor, /jf, the vibrating driving force
developed by the machine. B. Machine mounted upon an isolating system. In themechanical rectihneal system and mechanical network: m\, the mass of the machine.Cm/4 and 4rM , the compliance and mechanical rectilineal resistance of the four dampedspring mountings. z.vF, the mechanical rectilineal impedance of the floor. Jm, the vi-
brating driving force developed by the machine.
vibration may be so intense that it is intolerable. For these conditions
the machine may be isolated from the base or floor upon which it is
placed by introducing a mechanical isolating network.
A machine mounted directly on the floor is shown in Fig. W.SA.The mechanical rectilineal system and the mechanical network for verti-
cal vibrations is shown in Fig. W.SA. The driving force/i,/ is due to the
186 APPLICATIONS
vibrations of the machine. The mechanical network shows that the
only isolation in the system of Fig. 11.5^ is due to the mass of the
machine.
In the simple isolating system shown in Fig. 11.55 the machine is
mounted on damped springs. The compliance and the mechanical recti-
lineal resistance of the support is Cm and rm- Since there are four sup-
ports, these values become Cm/4 and Avm in the mechanical rectilineal
system and mechanical network for vertical vibrations. The mechanical
network depicts the action of the shunt circuit in reducing the force of
the vibration transmitted to the floor Zmp-
11.7. Mechanical Refrigerator Vibration Isolator
In the mechanical refrigerator a motor is used to drive a compressor.
Since the refrigerator is a home appliance it is important that the vibra-
LCFLOOR Zmf
BACK VIEW
MOTOR/ AND"COMPRES-SOR m, r
zl4
DAMPED Cm2SPRING ^
wr''^»^M
.RUBBER^FOOT MECHANICALCM2,r„2 RECTILINEAL
SYSTEMMECHANICALNETWORK
Fig. 11.6. Mechanical refrigerator vibration isolator. In the mechanical rectilineal sys-
tem and the mechanical network: zmf, the mechanical rectilineal impedance of the
floor. Cui/'i and 4rii/2, the compliance and mechanical rectilineal resistance of the four
rubber feet, mi, the mass of the case. Cmi/4 and 4rjif 2, the compliance and mechanical
rectihneal resistance of the four damped springs, mi, the mass of the motor and com-
pressor. Jm, the vibrating driving force developed by the machine.
tion and noise produced by the motor and compressor be as low in ampli-
tude as possible. These vibrations may be suppressed by the use of an
isolating mechanical network. The mechanical system, shown in Fig.
11.6, consists of the following elements: nii, the mass of the motor and
SHOCKPROOF INSTRUMENT MOUNTING 187
compressor. Cmi and rui, the compliance and the mechanical rectilineal
resistance of the springs and damping material entwined in the springs.
W2, the mass of the case. Cm2 and ^1-/2, the compliance and the mechan-
ical rectilineal resistance of the rubber feet, zmf, the mechanical rectilin-
eal resistance of the floor. Since there are four isolating supports for the
motor and compressor and four feet on the refrigerator, the elements in
the shunt circuits become Cmi/4 and ArMi for the isolating supports and
C?/if2/4 and 4rM2 for the rubber feet in the mechanical rectilineal system
and the mechanical network. The mechanical network illustrates howthe shunt circuit elements C,v/i/4, 4rAf 1 and Cm"2/4, 4rM2 reduce the force
delivered to the floor. The shunt circuit elements Cjv/i/4 and 4rMi also
reduces the force delivered to the case of the refrigerator and thereby
lessens the air-borne noises.
11.8. Shockproof Instrument Mounting
In order to obtain the maximum accuracy and reliability from gal-
vanometers and other similar instruments of high sensitivity it is neces-
sary that the mounting for the instrument be free from vibrations.
Very often these instruments must be used in buildings in which the
entire structure is vibrating. Any instrument support directly con-
nected to the building will vibrate and will in turn transmit this vibra-
tion to the instrument. Under these conditions the performance of the
instrument will be erratic. The instrument may be isolated from the
building vibrations by means of a mechanical network of the type shown
in Fig. 11.7. The instrument table legs are mounted on resiliant sup-
ports which are both a compliance Cmi and a resistance rjni- This sup-
port reduces the vibration of the table nti. The instrument is isolated
further by the compliance Cm2 and the mass ^2. A mechanical recti-
lineal resistance rM2 in the form of a dash pot is used to damp the vibra-
tions of the mass OT2. The driving force at each of the four legs is/vf.
Since there are four legs and four isolating supports, the elements in this
shunt circuit become Cm\/'^ and irux and the driving force becomes AJmin the mechanical rectilineal system and the mechanical network. Themechanical network illustrates the action of the vibrating system. Thevelocity of the mass ^2 is very small compared to the velocity of the
floor due to the series mass elements and shunt compliance and mechani-
cal rectilineal resistance elements. The mechanical network of Fig. 11.7
depicts the vertical modes of vibration. Of course, the system in Fig.
u APPLICATIONS
11.7 may vibrate in many other modes which may be solved by similar
analysis but, in general, the vertical motion is the most violent and
troublesome.
PERSPECTIVE VIEW
MECHANICALRECTILINEAL
SYSTEM
L-4f„-l
MECHANICALNETWORK
Fig. 11.7. Shockproof instrument mounting. In the mechanical rectilineal system andmechanical network: i/M, the combined driving force at the four legs. Cifi/4 and 4rM i,
the compliance and mechanical rectihneal resistance of the four feet. Cm2, the com-pliance of the spring suspension. r,v/2, the mechanical rectilineal resistance of the dash
pot. m-i, the mass of the instrument and carriage.
11.9. Automobile Suspension System
The riding qualities of an automobile depend primarily upon the
degree of isolation of the passenger from all types of vibration. One of
the principal sources of vibration is due to the uneven contour of the
road over which the automobile travels. The objective of automobile
designers is to reduce the vibration of the passenger to a practical mini-
mum. A schematic view of an automobile is shown in Fig. 11.8. This
system has many degrees of freedom, both rectilineal and rotational.
The system depicted in the mechanical rectilineal system and mechani-
cal network of Fig. 11.8 assumes that the forces at each of the four wheels
are equal in both amplitude and phase and that vibrations occur in a
vertical line. The vibrating system consists of the following elements:
/>/, the driving force at each tire. Cmi and tmi, the compliance and me-
chanical rectilineal resistance of the tires, mi, the mass of the tire, wheel
and axle. Cm2, the compliance of the spring. rM2i the mechanical recti-
lineal resistance of the shock absorber. m2, the mass of the frame, body,
engine, etc. Cms a-nd r.v/3, the compliance and mechanical rectilineal re-
AUTOMOBILE SUSPENSION SYSTEM 189
sistance of the cushion, m^, the mass of the passenger. Since there are
four tires, wheels, springs and shock absorbers, the elements corresponding
to these parts in the mechanical rectilineal system and mechanical net-
work are as follows: 4/3/, the driving force. CAfi/4 and 4r,/ifi the com-
pliance and mechanical rectilineal resistance of the tires. Ciif2/4, the
compliance of the springs. 4:rM2, the mechanical rectilineal resistance of
the shock absorbers. However, there is no change in the case of the
Qj""' I-Mr , 'mi
SCHEMATIC VIEW
MECHANICALRECTILINEAL
SYSTEMMECHANICALNETWORK
Fig. 11.8. Automobile suspension system. In the mechanical rectihneal system andmechanical network: 4/j/, the combined driving force at the four tires. Cmi/4 and
4r.\f 1, the compHance and mechanical rectihneal resistance of the four tires. 4?ni, the massof the four tires. Cji/2/4, the compliance of the four springs. 4rif 2, the mechanical rec-
tilineal resistance of the four shock absorbers, m^, the mass of the frame, body and
engine. Cms and r.i/a, the compHance and mechanical rectihneal resistance of the
cushion, mi, the mass of the passenger.
fn2. Cms, ^mz and W3 because these are single units in the schematic
view. The mechanical rectilineal resistance of the tires is quite small.
The mechanical rectilineal resistance of the springs is exceedingly small.
A low frequency oscillation with very little damping occurs due to the
resonance of the mass ^2 of the body with the compliances Cm2 and Cmiof the springs and tires. This oscillation is excited by a wavy road bed
and becomes very violent when the speed divided by the wavelength
corresponds to the resonant frequency. A high frequency oscillation
occurs due to the resonance of the mass »?2 of the wheels and axles with
the compliances Cmi arid Cm2 of the tires and springs. This oscillation
is excited by sharp discontinuities such as cobblestones. This resonance
190 APPLICATIONS
becomes so violent in the absence of damping that the wheels leave the
road. These uncontrolled oscillations require the introduction of some
form of damping for reducing the amplitude. A system has been de-
veloped in the form of the hydraulic shock absorber, which controls the
oscillations. It has been found that by the use of such damping means,
the compliance of the springs could be increased by use of "softer"
springs and the compliance of the tires increased by the use of balloon
tires. Both of these expedients have improved the riding qualities as
can be seen from a consideration of the mechanical network of Fig. 11.6.
A further improvement in riding qualities has been obtained by the use
of better designed cushions, that is, an appropriate ratio of compliance
Cm3 to mechanical rectilineal resistance tm^.
The above brief description illustrates how one of the vibration prob-
lems in an automobile may be solved by the use of analogies. As already
indicated, an automobile has several modes of vibration, both rectilineal
and rotational. For example, each wheel may be excited separately
which may introduce a rolling, pitching or swaying motion. All of these
may be analyzed by the use of analogies. The individual effects maybe superposed and the gross effect of all vibrations obtained. Most of
the forces, developed at the tires, are of the impulsive and not the sinus-
oidal type. In these cases the information on transients in electrical
circuits may be applied to the mechanical system as outlined in
Chapter VII.
Supplementary Note: Electrical, mechanical rectilineal, mechanical rotational
and acoustical ohms have been defined in the tables on pages 21, 22 and 23 andunder the definitions of these impedances. To avoid any possibility of ambi-guity, the explicit verbal definitions of these terms is listed below.
Electrical abohm.—An electrical resistance, reactance or impedance is said to
have a magnitude of one abohm when an electromotive force of one abvoltproduces a current of one abampere.Mechanical Ohm.—A mechanical rectilineal resistance, reactance or impedance
is said to have a magnitude of one mechanical ohm when a force of one dyneproduces a velocity of one centimeter per second.
Rotational Ohm.—A mechanical rotational resistance, reactance or impedanceis said to have a magnitude of one rotational ohm when a torque of one dynecentimeter produces an angular velocity of one radian per second.
Acoustical Ohm.—An acoustical resistance, reactance or impedance is said to
have a magnitude of one acoustical ohm when a pressure of one dyne per squarecentimeter produces a volume current of one cubic centimeter per second.
INDEX
Abampere, 6
Abvolt, 5
Acoustical,
capacitance, 10, 17, 18, 21, 23, 62, 76
impedance, 9, 23
narrow slit, 14
ohm, 9, 10, 23, 190
reactance, 10, 23
resistance, 10, 13, 23
system, 10, 25, 37
transformer, 88
wave filter, 92
Acoustomotive force, 6, 35
effective, 6
instantaneous, 6
maximum, 6
peak, 6
Angular velocity,
effective, 7
instantaneous, 7
maximum, 7
peak, 7
Applications, 180
Arbitrary force, 120
Automobile muffier, 180
Automobile suspension system, 188
Basic frequency, 4
Blocked electrical impedance, 126
Capacitance,
acoustical, 10, 17, 18, 21, 23, 62, 76
electrical, 8, 17, 18, 22, 23, 60, 74
Capacitive coupled systems and anal-
ogies, 45
Centimeter per second, 7
Clipper, electric, 182
Compliance, 9, 17, 18, 21, 22, 23, 60,
61,75
rectilineal, 9, 17, 21, 22, 60, 75
rotational, 9, 17, 18, 21, 23, 61, 75
Corrective networks, 52
resistance, 85
series, 86
shunt, 86
"T" type, 87
"tt" type, 87
series, 71
capacitance and analogies, 74
inductance and analogies, 72
inductance and capacitance in
parallel and analogies, 78
inductance and capacitance in
series and analogies, 76
resistance, inductance and capaci-
tance in parallel and analogies,
83
resistance, inductance and capaci-
tance in series and analogies, 80
shunt, 56
capacitance and analogies, 60
inductance and analogies, 58
inductance and capacitance in
parallel and analogies, 64
inductance and capacitance in
series and analogies, 62
resistance, inductance and capaci-
tance in parallel and analogies,
69
resistance, inductance and capaci-
tance in series and analogies,
67
Coupled systems and analogies, 45
Cubic centimeter per second, 7
191
192 INDEX
Current,
effective, 7
instantaneous, 6
maximum, 7
peak, 7
Cycle, 4
D'Alembert's principle, 33
Decibel, 11
Definitions, 4
Differential gear train, 55
Dimensions, 20, 21, 22, 23, 24
Dissipation, 29, 39
Driving systems, 124
electrodynamic, 124
electromagnetic, 126
electrostatic, 138
magnetostriction, 141
piezoelectric, 148
polarized balanced armature, 134
polarized reed armature, 130
unpolarized armature, 127
Duhamel's integral, 120
Dyne, 5
Dyne centimeter, 6
Dyne per square centimeter, 6
Effective,
acoustomotive force, 6
angular velocity, 7
current, 7
electromotive force, 5
force, 5
mechanomotive force, 5
rotatomotive force, 6
sound pressure, 6
torque, 6
velocity, 7
volume current, 8
Electrical,
abohm, 8, 22, 190
capacitance, 8, 17, 18, 22, 23, 60, 74
impedance, 8, 22
blocked, 126
normal, 126
Electrical, impedance (Cont.)
motional, 126
ohm, 8, 22
reactance, 8, 22
resistance, 8, 12, 22
system, 10, 25, 37
transformer, 88
wave filter, 92
Electric clipper, 182
Electrodynamic,
driving system, 124
generating system, 153
Electromagnetic,
driving system, 126
polarized balanced armature, 134
polarized reed armature, 130
unpolarized armature, 127
generating system, 155
balanced armature, 157
reed armature, 155
Electromotive force, 5, 22, 33
effective, 5
instantaneous, 5
maximum, 5
peak, 5
Electrostatic,
driving system, 138
generating system, 158
Element, 10, 12, 19, 21, 22, 23
acoustical, 13, 16, 18, 19, 21, 23
electrical, 12, 15, 17, 19, 21, 22
mechanical rectilineal, 13, 15, 17, 21,
22
mechanical rotational, 13, 15, 18,
21,23Energy,
kinetic, 27, 38
potential, 28, 39
Epicyclic gear train, 54
Equation, Lagrange's, 40
Equation of motion, 30, 40
Filters, wave (see wave filters)
Force, 5, 21, 22
acoustomotive, 6, 35
INDEX 193
Force (Cont.)
arbitrary, 120
effective, 5
electromotive, 5, 21, 33
instantaneous, S
maximum, 5
mechanomotive, 5, 34
peak, 5
rotatomotive, 6, 33
Frequency, 4
basic, 4fundamental, 4
resonant, 32
Gear train, 54, S5
Generating systems, 153
electrodynamic, 153
electromagnetic, 155
balanced armature, 157
reed armature, 155
electrostatic, 158
magnetostriction, 162
piezoelectric, 165
Harmonics, 4
Heaviside Operational Calculus, 106
Impedance,acoustical, 9, 23
electrical, 8, 22
blocked, 126
normal, 126
motional, 126
mechanical, 8, 22, 23
mechanical rectilineal, 8, 22
mechanical rotational, 9, 23
parallel, 52
rotational, 9, 23
Inductance, 8, 15, 21, 22, 58, 72
Inductive coupled systems and anal-
ogies, 46
Inertance, 10, 15, 16, 21, 23, 60, 74
Inertia, moment of, 9, 15, 21, 23, 59, 73
Instantaneous,
acoustomotive force, 6
Instantaneous (Cont.)
electromotive force, 5
force, 5
mechanomotive force, 5
rotatomotive force, 6
sound pressure, 6
torque, 6
velocity, 7
volume current, 8
Instrument, shock proof mounting, 187
Integral, Duhamel's, 120
Introduction, 1
Joule effect, magnetostriction, 142
Kinetic energy, 27, 38
Kirchhoff's law, 33
Lagrange's equations, 40
Law, Kirchhoff's, 33
Line, 52, 56, 71
Loud speaker, 183
Machine vibration isolator, 185
Magnetostriction,
driving system, 141
generating system, 162
Mass, 9, 15, 21, 22, 58, 73
Maximum,acoustomotive force, 6
angular velocity, 7
current, 7
electromotive force, 5
force, 34
mechanomotive force, 5
rotatomotive force, 6
sound pressure, 6
torque, 6
velocity, 7
volume current, 8
Mechanical,
impedance, 8, 22, 23
ohm, 8, 9, 22, 190
reactance, 9, 22, 23
rectilineal impedance, 8, 22
194 INDEX
Mechanical (Cont.)
rectilineal reactance, 9, 22
rectilineal resistance, 9, 13, 22
rectilineal system, 10, 41
rectilineal wave filters, 92
resistance, 9, 13, 22, 23
rotational impedance, 9, 23
rotational reactance, 9, 23
rotational resistance, 9, 13, 23
rotational system, 10, 42
rotational wave filters, 92
transformer, 88
Mechanical refrigerator vibration iso-
lator, 186
Moment of inertia, 9, IS, 21, 23, 59,
73
Motional electrical impedance, 125,
126
Muffler, automobile, 180
Narrow slit, 14
Networks, 2, 52, 125, 181 (see cor-
rective networks)
acoustical, 52, 125, 181
electrical, 52, 125, 181
mechanical rectilineal, 52, 125, 181
mechanical rotational, 52, 125, 181
resistance corrective, 85
series corrective, 71
shunt corrective, 56
Normal electrical impedance, 126
Octave, 4
Ohm,acoustical, 9, 10, 23, 190
electrical, 8, 22, 190
mechanical, 8, 9, 22, 190
rotational, 9, 23, 190
One degree of freedom, 25
acoustical, 25
electrical, 25
mechanical rectilineal, 25
mechanical rotatomotive, 25
Operational Calculus, Heaviside,
106
Parallel,
impedances, 52, 64, 69
acoustical, 52, 64, 69
electrical, 52, 64, 69
mechanical rectilineal, 52, 64, 69
mechanical rotatomotive, 52, 64,
69
Peak,
acoustomotive force, 6
angular velocity, 7
current, 7
electromotive force, 5
force, 5
mechanomotive force, 5
rotatomotive force, 6
sound pressure, 6
torque, 6
velocity, 7
volume current, 8
Period, 4
Periodic quantity, 4
Piezoelectric,
driving system, 148
generating system, 165
"it" type network, 87
Planetary gear train, 54
Potential energy, 28, 39
Pressure,
sound, 6, 21, 23
static, 6
Quartz crystal, 148, 165
Radians per second, 7
Reactance,
acoustical, 10, 23
electrical, 8, 22
mechanical, 9, 22, 23
mechanical rectilineal, 9, 22
mechanical rotational, 9, 23
rotational, 9
Reciprocity theorems (see theorems)
Resistance,
acoustical, 10, 13, 23
electrical, 8, 12, 22
INDEX 195
Resistance (Cont.)
mechanical, 9, 13, 22, 23
mechanical rectilineal, 9, 13, 22
mechanical rotational, 9, 13, 22
networlcs, 85
rotational, 9, 13, 23
Resonant frequency, 32
Rotational,
compliance, 9, 17, 18, 21, 23, 61, 75
impedance, 9, 23
ohm, 9, 23, 190
reactance, 9
resistance, 9, 13, 23
Rotational vibration damper, 184
Rotatomotive force, 6, 35
effective, 6
instantaneous, 6
maximum, 6
peak, 6
Series corrective networks (see cor-
rective networks)
Shock proof instrument mounting, 187
Shunt corrective networks (see cor-
rective networks)
Slit, 14
Sound pressure, 6, 21, 23
effective, 6
instantaneous, 6
maximum, 6
peak, 6
Static pressure, 6
Stiffness, 18, 28
Subharmonic, 5
Superposition theorem, 179
Suspension systems, automobile, 188
Systems,
acoustical, 10, 42
coupled, 45
electrical, 10, 41
mechanical rectilineal, 10, 41
mechanical rotational, 10, 42
one degree of freedom, 25
three degrees of freedom, 48
two degrees of freedom, 37
Table, 21, 22, 23
Theorems,
reciprocity, 171
acoustical, 173
acoustical-mechanical-electrical-
mechanical-acoustical, 178
electrical, 171
electrical-mechanical, 176
electrical-mechanical-acoustical,
177
electrical-mechanical-acoustical-
mechanical-electrical, 177
mechanical-acoustical, 175
mechanical rectilineal, 172
mechanical rotational, 173
superposition, 179
Thevenin's, 178
acoustical, 178
electrical, 178
mechanical rectilineal, 178
mechanical rotational, 178
Three degrees of freedom, 37, 45
Torque, 6, 21, 23
effective, 6
instantaneous, 6
maximum, 6
peak, 6
Transducer, 11
Transformer,
acoustical, 88
electrical, 88
mechanical rectilineal, 88
mechanical rotational, 88
Transient response,
electrical resistance and electrical
capacitance in series and anal-
ogies, 111
electrical resistance, inductance andelectrical capacitance in series
and analogies, 114
inductance and electrical resistance
in series and analogies, 107
Transients, 105
arbitrary force, 120
Duhamel's integral, 122
196 INDEX
Transients (Cont.)
Heaviside's Calculus, 106
unit function, 106
Transmission,
gain, 11
loss, 11
"T" type network, 87
Two degrees of freedom, 37, 45
Unit function, 106
Units, 20, 21, 22, 23, 24
Velocity, 7, 22
angular, 7, 23
effective, 7
instantaneous, 7
linear, 7, 22
maximum, 7
peak, 7
Vibration,
machine isolator, 185
mechanical refrigerator isolator, 186
rotational damper, 184
Villari effect, magnetostriction, 143,
163
Volume current, 8, 23
effective, 8
instantaneous, 8
maximum, 8
peak, 8
Wave, 5
Wave filters,
band elimination, 92, 93, 101
band pass, 92, 93, 97
high pass, 92, 93, 95
low pass, 92, 93, 94
response characteristics, 93
Wavelength, 5
"X" cut quartz crystal, 148
Young's modulus, 144, 146, 147, 149,
150, 151, 163, 165, 167